Coverage for local/lib/python2.7/site-packages/sage/rings/number_field/number

def NumberField(polynomial, name=None, check=True, names=None, embedding=None, latex_name=None, assume_disc_small=False, maximize_at_primes=None, structure=None):

r"""

Return *the* number field (or tower of number fields) defined by the

irreducible ``polynomial``.

INPUT:

- ``polynomial`` - a polynomial over `\QQ` or a number field, or a list

of such polynomials.

- ``name`` - a string or a list of strings, the names of the generators

- ``check`` - a boolean (default: ``True``); do type checking and

irreducibility checking.

- ``embedding`` - ``None``, an element, or a list of elements, the

images of the generators in an ambient field (default: ``None``)

- ``latex_name`` - ``None``, a string, or a list of strings (default:

``None``), how the generators are printed for latex output

- ``assume_disc_small`` -- a boolean (default: ``False``); if ``True``,

assume that no square of a prime greater than PARI's primelimit

(which should be 500000); only applies for absolute fields at

present.

- ``maximize_at_primes`` -- ``None`` or a list of primes (default:

``None``); if not ``None``, then the maximal order is computed by

maximizing only at the primes in this list, which completely avoids

having to factor the discriminant, but of course can lead to wrong

results; only applies for absolute fields at present.

- ``structure`` -- ``None``, a list or an instance of

:class:`structure.NumberFieldStructure` (default: ``None``),

internally used to pass in additional structural information, e.g.,

about the field from which this field is created as a subfield.

EXAMPLES::

sage: z = QQ['z'].0

sage: K = NumberField(z^2 - 2,'s'); K

Number Field in s with defining polynomial z^2 - 2

sage: s = K.0; s

sage: s*s

sage: s^2

Constructing a relative number field::

sage: K.<a> = NumberField(x^2 - 2)

sage: R.<t> = K[]

sage: L. = K.extension(t^3+t+a); L

Number Field in b with defining polynomial t^3 + t + a over its base field

sage: L.absolute_field('c')

Number Field in c with defining polynomial x^6 + 2*x^4 + x^2 - 2

sage: a*b

a*b

sage: L(a)

sage: L.lift_to_base(b^3 + b)

-a

Constructing another number field::

sage: k. = NumberField(x^2 + 1)

sage: R.<z> = k[]

sage: m.<j> = NumberField(z^3 + i*z + 3)

sage: m

Number Field in j with defining polynomial z^3 + i*z + 3 over its base field

Number fields are globally unique::

sage: K.<a> = NumberField(x^3 - 5)

sage: a^3

sage: L.<a> = NumberField(x^3 - 5)

sage: K is L

True

Equality of number fields depends on the variable name of the

defining polynomial::

sage: x = polygen(QQ, 'x'); y = polygen(QQ, 'y')

sage: k.<a> = NumberField(x^2 + 3)

sage: m.<a> = NumberField(y^2 + 3)

sage: k

Number Field in a with defining polynomial x^2 + 3

sage: m

Number Field in a with defining polynomial y^2 + 3

sage: k == m

False

In case of conflict of the generator name with the name given by the preparser, the name given by the preparser takes precedence::

sage: K. = NumberField(x^2 + 5, 'a'); K

Number Field in b with defining polynomial x^2 + 5

One can also define number fields with specified embeddings, may be used

for arithmetic and deduce relations with other number fields which would

not be valid for an abstract number field. ::

sage: K.<a> = NumberField(x^3-2, embedding=1.2)

sage: RR.coerce_map_from(K)

Composite map:

From: Number Field in a with defining polynomial x^3 - 2

To: Real Field with 53 bits of precision

Defn: Generic morphism:

From: Number Field in a with defining polynomial x^3 - 2

To: Real Lazy Field

Defn: a -> 1.259921049894873?

then

Conversion via _mpfr_ method map:

From: Real Lazy Field

To: Real Field with 53 bits of precision

sage: RR(a)

1.25992104989487

sage: 1.1 + a

2.35992104989487

sage: b = 1/(a+1); b

1/3*a^2 - 1/3*a + 1/3

sage: RR(b)

0.442493334024442

sage: L. = NumberField(x^6-2, embedding=1.1)

sage: L(a)

b^2

sage: a + b

b^2 + b

Note that the image only needs to be specified to enough precision

to distinguish roots, and is exactly computed to any needed

precision::

sage: RealField(200)(a)

1.2599210498948731647672106072782283505702514647015079800820

One can embed into any other field::

sage: K.<a> = NumberField(x^3-2, embedding=CC.gen()-0.6)

sage: CC(a)

-0.629960524947436 + 1.09112363597172*I

sage: L = Qp(5)

sage: f = polygen(L)^3 - 2

sage: K.<a> = NumberField(x^3-2, embedding=f.roots()[0][0])

sage: a + L(1)

4 + 2*5^2 + 2*5^3 + 3*5^4 + 5^5 + 4*5^6 + 2*5^8 + 3*5^9 + 4*5^12 + 4*5^14 + 4*5^15 + 3*5^16 + 5^17 + 5^18 + 2*5^19 + O(5^20)

sage: L. = NumberField(x^6-x^2+1/10, embedding=1)

sage: K.<a> = NumberField(x^3-x+1/10, embedding=b^2)

sage: a+b

b^2 + b

sage: CC(a) == CC(b)^2

True

sage: K.coerce_embedding()

Generic morphism:

From: Number Field in a with defining polynomial x^3 - x + 1/10

To: Number Field in b with defining polynomial x^6 - x^2 + 1/10

Defn: a -> b^2

The ``QuadraticField`` and ``CyclotomicField`` constructors

create an embedding by default unless otherwise specified::

sage: K.<zeta> = CyclotomicField(15)

sage: CC(zeta)

0.913545457642601 + 0.406736643075800*I

sage: L.<sqrtn3> = QuadraticField(-3)

sage: K(sqrtn3)

2*zeta^5 + 1

sage: sqrtn3 + zeta

2*zeta^5 + zeta + 1

Comparison depends on the (real) embedding specified (or the one selected by default).

Note that the codomain of the embedding must be `QQbar` or `AA` for this to work

(see :trac:`20184`)::

sage: N.<g> = NumberField(x^3+2,embedding=1)

sage: 1 < g

False

sage: g > 1

False

sage: RR(g)

-1.25992104989487

If no embedding is specified or is complex, the comparison is not returning something

meaningful.::

sage: N.<g> = NumberField(x^3+2)

sage: 1 < g

False

sage: g > 1

True

Since SageMath 6.9, number fields may be defined by polynomials

that are not necessarily integral or monic. The only notable

practical point is that in the PARI interface, a monic integral

polynomial defining the same number field is computed and used::

sage: K.<a> = NumberField(2*x^3 + x + 1)

sage: K.pari_polynomial()

x^3 - x^2 - 2

Elements and ideals may be converted to and from PARI as follows::

sage: pari(a)

Mod(-1/2*y^2 + 1/2*y, y^3 - y^2 - 2)

sage: K(pari(a))

sage: I = K.ideal(a); I

Fractional ideal (a)

sage: I.pari_hnf()

[1, 0, 0; 0, 1, 0; 0, 0, 1/2]

sage: K.ideal(I.pari_hnf())

Fractional ideal (a)

Here is an example where the field has non-trivial class group::

sage: L. = NumberField(3*x^2 - 1/5)

sage: L.pari_polynomial()

x^2 - 15

sage: J = L.primes_above(2)[0]; J

Fractional ideal (2, 15*b + 1)

sage: J.pari_hnf()

[2, 1; 0, 1]

sage: L.ideal(J.pari_hnf())

Fractional ideal (2, 15*b + 1)

An example involving a variable name that defines a function in

PARI::

sage: theta = polygen(QQ, 'theta')

sage: M.<z> = NumberField([theta^3 + 4, theta^2 + 3]); M

Number Field in z0 with defining polynomial theta^3 + 4 over its base field

TESTS::

sage: x = QQ['x'].gen()

sage: y = ZZ['y'].gen()

sage: K = NumberField(x^3 + x + 3, 'a'); K

Number Field in a with defining polynomial x^3 + x + 3

sage: K.defining_polynomial().parent()

Univariate Polynomial Ring in x over Rational Field

sage: L = NumberField(y^3 + y + 3, 'a'); L

Number Field in a with defining polynomial y^3 + y + 3

sage: L.defining_polynomial().parent()

Univariate Polynomial Ring in y over Rational Field

sage: W1 = NumberField(x^2+1,'a')

sage: K.<x> = CyclotomicField(5)[]

sage: W.<a> = NumberField(x^2 + 1); W

Number Field in a with defining polynomial x^2 + 1 over its base field

The following has been fixed in :trac:`8800`::

sage: P.<x> = QQ[]

sage: K.<a> = NumberField(x^3-5,embedding=0)

sage: L. = K.extension(x^2+a)

sage: F, R = L.construction()

sage: F(R) == L # indirect doctest

True

Check that :trac:`11670` has been fixed::

sage: K.<a> = NumberField(x^2 - x - 1)

sage: loads(dumps(K)) is K

True

sage: K.<a> = NumberField(x^3 - x - 1)

sage: loads(dumps(K)) is K

True

sage: K.<a> = CyclotomicField(7)

sage: loads(dumps(K)) is K

True

Another problem that was found while working on :trac:`11670`,

``maximize_at_primes`` and ``assume_disc_small`` were lost when pickling::

sage: K.<a> = NumberField(x^3-2, assume_disc_small=True, maximize_at_primes=[2], latex_name='\\alpha', embedding=2^(1/3))

sage: L = loads(dumps(K))

sage: L._assume_disc_small

True

sage: L._maximize_at_primes

(2,)

It is an error not to specify the generator::

sage: K = NumberField(x^2-2)

Traceback (most recent call last):

...

TypeError: You must specify the name of the generator.

Check that we can construct morphisms to matrix space (:trac:`23418`)::

sage: t = polygen(QQ)

sage: K = NumberField(t^4 - 2, 'a')

sage: K.hom([K.gen().matrix()])

Ring morphism:

From: Number Field in a with defining polynomial x^4 - 2

To: Full MatrixSpace of 4 by 4 dense matrices over Rational Field

Defn: a |--> [0 1 0 0]

[0 0 1 0]

[0 0 0 1]

[2 0 0 0]

"""

if names is not None:

name = names

if isinstance(polynomial, (list,tuple)):

return NumberFieldTower(polynomial, names=name, check=check, embeddings=embedding, latex_names=latex_name, assume_disc_small=assume_disc_small, maximize_at_primes=maximize_at_primes, structures=structure)

return NumberField_version2(polynomial=polynomial, name=name, check=check, embedding=embedding, latex_name=latex_name, assume_disc_small=assume_disc_small, maximize_at_primes=maximize_at_primes, structure=structure)

class NumberFieldFactory(UniqueFactory):

r"""

Factory for number fields.

This should usually not be called directly, use :meth:`NumberField`

instead.

INPUT:

- ``polynomial`` - a polynomial over `\QQ` or a number field.

- ``name`` - a string (default: ``'a'``), the name of the generator

- ``check`` - a boolean (default: ``True``); do type checking and

irreducibility checking.

- ``embedding`` - ``None`` or an element, the images of the generator

in an ambient field (default: ``None``)

- ``latex_name`` - ``None`` or a string (default: ``None``), how the

generator is printed for latex output

- ``assume_disc_small`` -- a boolean (default: ``False``); if ``True``,

assume that no square of a prime greater than PARI's primelimit

(which should be 500000); only applies for absolute fields at

present.

- ``maximize_at_primes`` -- ``None`` or a list of primes (default:

``None``); if not ``None``, then the maximal order is computed by

maximizing only at the primes in this list, which completely avoids

having to factor the discriminant, but of course can lead to wrong

results; only applies for absolute fields at present.

- ``structure`` -- ``None`` or an instance of

:class:`structure.NumberFieldStructure` (default: ``None``),

internally used to pass in additional structural information, e.g.,

about the field from which this field is created as a subfield.

TESTS::

sage: from sage.rings.number_field.number_field import NumberFieldFactory

sage: nff = NumberFieldFactory("number_field_factory")

sage: R.<x> = QQ[]

sage: nff(x^2 + 1, name='a', check=False, embedding=None, latex_name=None, assume_disc_small=False, maximize_at_primes=None, structure=None)

Number Field in a with defining polynomial x^2 + 1

Pickling preserves the ``structure()`` of a number field::

sage: K.<a> = QuadraticField(2)

sage: L. = K.change_names()

sage: M = loads(dumps(L))

sage: M.structure()

(Isomorphism given by variable name change map:

From: Number Field in b with defining polynomial x^2 - 2

To: Number Field in a with defining polynomial x^2 - 2,

Isomorphism given by variable name change map:

From: Number Field in a with defining polynomial x^2 - 2

To: Number Field in b with defining polynomial x^2 - 2)

"""

def create_key_and_extra_args(self, polynomial, name, check, embedding, latex_name, assume_disc_small, maximize_at_primes, structure):

r"""

Create a unique key for the number field specified by the parameters.

TESTS::

sage: from sage.rings.number_field.number_field import NumberFieldFactory

sage: nff = NumberFieldFactory("number_field_factory")

sage: R.<x> = QQ[]

sage: nff.create_key_and_extra_args(x^2+1, name='a', check=False, embedding=None, latex_name=None, assume_disc_small=False, maximize_at_primes=None, structure=None)

((Rational Field, x^2 + 1, ('a',), None, 'a', None, False, None),

{'check': False})

"""

if name is None:

raise TypeError("You must specify the name of the generator.")

name = normalize_names(1, name)

if not is_Polynomial(polynomial):

try:

polynomial = polynomial.polynomial(QQ)

except (AttributeError, TypeError):

raise TypeError("polynomial (=%s) must be a polynomial." % polynomial)

# convert polynomial to a polynomial over a field

polynomial = polynomial.change_ring(polynomial.base_ring().fraction_field())

# normalize embedding

if isinstance(embedding, (list,tuple)):

if len(embedding) != 1:

raise TypeError("embedding must be a list of length 1")

embedding = embedding[0]

if embedding is not None:

x = number_field_morphisms.root_from_approx(polynomial, embedding)

embedding = (x.parent(), x)

# normalize latex_name

if isinstance(latex_name, (list, tuple)):

if len(latex_name) != 1:

raise TypeError("latex_name must be a list of length 1")

latex_name = latex_name[0]

if latex_name is None:

latex_name = latex_variable_name(name[0])

if maximize_at_primes is not None:

maximize_at_primes = tuple(maximize_at_primes)

# normalize structure

if isinstance(structure, (list, tuple)):

if len(structure) != 1:

raise TypeError("structure must be a list of length 1")

structure = structure[0]

return (polynomial.base_ring(), polynomial, name, embedding, latex_name, maximize_at_primes, assume_disc_small, structure), {"check":check}

def create_object(self, version, key, check):

r"""

Create the unique number field defined by ``key``.

TESTS::

sage: from sage.rings.number_field.number_field import NumberFieldFactory

sage: nff = NumberFieldFactory("number_field_factory")

sage: R.<x> = QQ[]

sage: nff.create_object(None, (QQ, x^2 + 1, ('a',), None, None, None, False, None), check=False)

Number Field in a with defining polynomial x^2 + 1

"""

base, polynomial, name, embedding, latex_name, maximize_at_primes, assume_disc_small, structure = key

if isinstance(base, NumberField_generic):

from sage.rings.number_field.number_field_rel import NumberField_relative

# Relative number fields do not support embeddings.

return NumberField_relative(base, polynomial, name[0], latex_name,

check=check, embedding=None,

structure=structure)

if polynomial.degree() == 2:

return NumberField_quadratic(polynomial, name, latex_name, check, embedding, assume_disc_small=assume_disc_small, maximize_at_primes=maximize_at_primes, structure=structure)

else:

return NumberField_absolute(polynomial, name, latex_name, check, embedding, assume_disc_small=assume_disc_small, maximize_at_primes=maximize_at_primes, structure=structure)

NumberField_version2 = NumberFieldFactory("sage.rings.number_field.number_field.NumberField_version2")

def NumberFieldTower(polynomials, names, check=True, embeddings=None, latex_names=None, assume_disc_small=False, maximize_at_primes=None, structures=None):

"""

Create the tower of number fields defined by the polynomials in the list

``polynomials``.

INPUT:

- ``polynomials`` - a list of polynomials. Each entry must be polynomial

which is irreducible over the number field generated by the roots of the

following entries.

- ``names`` - a list of strings or a string, the names of the generators of

the relative number fields. If a single string, then names are generated

from that string.

- ``check`` - a boolean (default: ``True``), whether to check that the

polynomials are irreducible

- ``embeddings`` - a list of elements or ``None`` (default: ``None``),

embeddings of the relative number fields in an ambient field.

- ``latex_names`` - a list of strings or ``None`` (default: ``None``), names

used to print the generators for latex output.

- ``assume_disc_small`` -- a boolean (default: ``False``); if ``True``,

assume that no square of a prime greater than PARI's primelimit

(which should be 500000); only applies for absolute fields at

present.

- ``maximize_at_primes`` -- ``None`` or a list of primes (default:

``None``); if not ``None``, then the maximal order is computed by

maximizing only at the primes in this list, which completely avoids

having to factor the discriminant, but of course can lead to wrong

results; only applies for absolute fields at present.

- ``structures`` -- ``None`` or a list (default: ``None``), internally used

to provide additional information about the number field such as the

field from which it was created.

OUTPUT:

Returns the relative number field generated by a root of the first entry of

``polynomials`` over the relative number field generated by root of the

second entry of ``polynomials`` ... over the number field over which the

last entry of ``polynomials`` is defined.

EXAMPLES::

sage: k.<a,b,c> = NumberField([x^2 + 1, x^2 + 3, x^2 + 5]); k # indirect doctest

Number Field in a with defining polynomial x^2 + 1 over its base field

sage: a^2

-1

sage: b^2

-3

sage: c^2

-5

sage: (a+b+c)^2

(2*b + 2*c)*a + 2*c*b - 9

The Galois group is a product of 3 groups of order 2::

sage: k.galois_group(type="pari")

Galois group PARI group [8, 1, 3, "E(8)=2[x]2[x]2"] of degree 8 of the Number Field in a with defining polynomial x^2 + 1 over its base field

Repeatedly calling base_field allows us to descend the internally

constructed tower of fields::

sage: k.base_field()

Number Field in b with defining polynomial x^2 + 3 over its base field

sage: k.base_field().base_field()

Number Field in c with defining polynomial x^2 + 5

sage: k.base_field().base_field().base_field()

Rational Field

In the following example the second polynomial is reducible over

the first, so we get an error::

sage: v = NumberField([x^3 - 2, x^3 - 2], names='a')

Traceback (most recent call last):

...

ValueError: defining polynomial (x^3 - 2) must be irreducible

We mix polynomial parent rings::

sage: k.<y> = QQ[]

sage: m = NumberField([y^3 - 3, x^2 + x + 1, y^3 + 2], 'beta')

sage: m

Number Field in beta0 with defining polynomial y^3 - 3 over its base field

sage: m.base_field ()

Number Field in beta1 with defining polynomial x^2 + x + 1 over its base field

A tower of quadratic fields::

sage: K.<a> = NumberField([x^2 + 3, x^2 + 2, x^2 + 1])

sage: K

Number Field in a0 with defining polynomial x^2 + 3 over its base field

sage: K.base_field()

Number Field in a1 with defining polynomial x^2 + 2 over its base field

sage: K.base_field().base_field()

Number Field in a2 with defining polynomial x^2 + 1

A bigger tower of quadratic fields::

sage: K.<a2,a3,a5,a7> = NumberField([x^2 + p for p in [2,3,5,7]]); K

Number Field in a2 with defining polynomial x^2 + 2 over its base field

sage: a2^2

-2

sage: a3^2

-3

sage: (a2+a3+a5+a7)^3

((6*a5 + 6*a7)*a3 + 6*a7*a5 - 47)*a2 + (6*a7*a5 - 45)*a3 - 41*a5 - 37*a7

The function can also be called by name::

sage: NumberFieldTower([x^2 + 1, x^2 + 2], ['a','b'])

Number Field in a with defining polynomial x^2 + 1 over its base field

"""

try:

names = normalize_names(len(polynomials), names)

except IndexError:

names = normalize_names(1, names)

if len(polynomials) > 1:

names = ['%s%s'%(names[0], i) for i in range(len(polynomials))]

if embeddings is None:

embeddings = [None] * len(polynomials)

if latex_names is None:

latex_names = [None] * len(polynomials)

if structures is None:

structures = [None] * len(polynomials)

if not isinstance(polynomials, (list, tuple)):

raise TypeError("polynomials must be a list or tuple")

if len(polynomials) == 0:

return QQ

if len(polynomials) == 1:

return NumberField(polynomials[0], names=names, check=check, embedding=embeddings[0], latex_name=latex_names[0], assume_disc_small=assume_disc_small, maximize_at_primes=maximize_at_primes, structure=structures[0])

# create the relative number field defined by f over the tower defined by polynomials[1:]

f = polynomials[0]

name = names[0]

w = NumberFieldTower(polynomials[1:], names=names[1:], check=check, embeddings=embeddings[1:], latex_names=latex_names[1:], assume_disc_small=assume_disc_small, maximize_at_primes=maximize_at_primes, structures=structures[1:])

var = f.variable_name() if is_Polynomial(f) else 'x'

R = w[var] # polynomial ring

return w.extension(R(f), name, check=check, embedding=embeddings[0], structure=structures[0]) # currently, extension does not accept assume_disc_small, or maximize_at_primes

def QuadraticField(D, name='a', check=True, embedding=True, latex_name='sqrt', **args):

r"""

Return a quadratic field obtained by adjoining a square root of

`D` to the rational numbers, where `D` is not a

perfect square.

INPUT:

- ``D`` - a rational number

- ``name`` - variable name (default: 'a')

- ``check`` - bool (default: True)

- ``embedding`` - bool or square root of D in an

ambient field (default: True)

- ``latex_name`` - latex variable name (default: \sqrt{D})

OUTPUT: A number field defined by a quadratic polynomial. Unless

otherwise specified, it has an embedding into `\RR` or

`\CC` by sending the generator to the positive

or upper-half-plane root.

EXAMPLES::

sage: QuadraticField(3, 'a')

Number Field in a with defining polynomial x^2 - 3

sage: K.<theta> = QuadraticField(3); K

Number Field in theta with defining polynomial x^2 - 3

sage: RR(theta)

1.73205080756888

sage: QuadraticField(9, 'a')

Traceback (most recent call last):

...

ValueError: D must not be a perfect square.

sage: QuadraticField(9, 'a', check=False)

Number Field in a with defining polynomial x^2 - 9

Quadratic number fields derive from general number fields.

sage: from sage.rings.number_field.number_field import is_NumberField

sage: type(K)

sage: is_NumberField(K)

True

Quadratic number fields are cached::

sage: QuadraticField(-11, 'a') is QuadraticField(-11, 'a')

True

By default, quadratic fields come with a nice latex representation::

sage: K.<a> = QuadraticField(-7)

sage: latex(K)

\Bold{Q}(\sqrt{-7})

sage: latex(a)

\sqrt{-7}

sage: latex(1/(1+a))

-\frac{1}{8} \sqrt{-7} + \frac{1}{8}

sage: K.latex_variable_name()

'\\sqrt{-7}'

We can provide our own name as well::

sage: K.<a> = QuadraticField(next_prime(10^10), latex_name=r'\sqrt{D}')

sage: 1+a

a + 1

sage: latex(1+a)

\sqrt{D} + 1

sage: latex(QuadraticField(-1, 'a', latex_name=None).gen())

The name of the generator does not interfere with Sage preparser, see :trac:`1135`::

sage: K1 = QuadraticField(5, 'x')

sage: K2.<x> = QuadraticField(5)

sage: K3.<x> = QuadraticField(5, 'x')

sage: K1 is K2

True

sage: K1 is K3

True

sage: K1

Number Field in x with defining polynomial x^2 - 5

Note that, in presence of two different names for the generator,

the name given by the preparser takes precedence::

sage: K4.<y> = QuadraticField(5, 'x'); K4

Number Field in y with defining polynomial x^2 - 5

sage: K1 == K4

False

TESTS::

sage: QuadraticField(-11, 'a') is QuadraticField(-11, 'a', latex_name='Z')

False

sage: QuadraticField(-11, 'a') is QuadraticField(-11, 'a', latex_name=None)

False

"""

D = QQ(D)

if check:

if D.is_square():

raise ValueError("D must not be a perfect square.")

R = QQ['x']

f = R([-D, 0, 1])

if embedding is True:

if D > 0:

embedding = RLF(D).sqrt()

else:

embedding = CLF(D).sqrt()

if latex_name == 'sqrt':

latex_name = r'\sqrt{%s}' % D

return NumberField(f, name, check=False, embedding=embedding, latex_name=latex_name, **args)

def is_AbsoluteNumberField(x):

"""

Return True if x is an absolute number field.

EXAMPLES::

sage: from sage.rings.number_field.number_field import is_AbsoluteNumberField

sage: is_AbsoluteNumberField(NumberField(x^2+1,'a'))

True

sage: is_AbsoluteNumberField(NumberField([x^3 + 17, x^2+1],'a'))

False

The rationals are a number field, but they're not of the absolute

number field class.

sage: is_AbsoluteNumberField(QQ)

False

"""

return isinstance(x, NumberField_absolute)

def is_QuadraticField(x):

r"""

Return True if x is of the quadratic *number* field type.

EXAMPLES::

sage: from sage.rings.number_field.number_field import is_QuadraticField

sage: is_QuadraticField(QuadraticField(5,'a'))

True

sage: is_QuadraticField(NumberField(x^2 - 5, 'b'))

True

sage: is_QuadraticField(NumberField(x^3 - 5, 'b'))

False

A quadratic field specially refers to a number field, not a finite

field::

sage: is_QuadraticField(GF(9,'a'))

False

"""

return isinstance(x, NumberField_quadratic)

class CyclotomicFieldFactory(UniqueFactory):

r"""

Return the `n`-th cyclotomic field, where n is a positive integer,

or the universal cyclotomic field if ``n==0``.

For the documentation of the universal cyclotomic field, see

:class:`~sage.rings.universal_cyclotomic_field.UniversalCyclotomicField`.

INPUT:

- ``n`` - a nonnegative integer, default:``0``

- ``names`` - name of generator (optional - defaults to zetan)

- ``bracket`` - Defines the brackets in the case of ``n==0``, and

is ignored otherwise. Can be any even length string, with ``"()"`` being the default.

- ``embedding`` - bool or n-th root of unity in an

ambient field (default True)

EXAMPLES:

If called without a parameter, we get the :class:`universal cyclotomic

field<sage.rings.universal_cyclotomic_field.UniversalCyclotomicField>`::

sage: CyclotomicField()

Universal Cyclotomic Field

We create the `7`\th cyclotomic field

`\QQ(\zeta_7)` with the default generator name.

sage: k = CyclotomicField(7); k

Cyclotomic Field of order 7 and degree 6

sage: k.gen()

zeta7

The default embedding sends the generator to the complex primitive

`n^{th}` root of unity of least argument.

sage: CC(k.gen())

0.623489801858734 + 0.781831482468030*I

Cyclotomic fields are of a special type.

sage: type(k)

We can specify a different generator name as follows.

sage: k.<z7> = CyclotomicField(7); k

Cyclotomic Field of order 7 and degree 6

sage: k.gen()

The `n` must be an integer.

sage: CyclotomicField(3/2)

Traceback (most recent call last):

...

TypeError: no conversion of this rational to integer

The degree must be nonnegative.

sage: CyclotomicField(-1)

Traceback (most recent call last):

...

ValueError: n (=-1) must be a positive integer

The special case `n=1` does *not* return the rational

numbers::

sage: CyclotomicField(1)

Cyclotomic Field of order 1 and degree 1

Due to their default embedding into `\CC`,

cyclotomic number fields are all compatible.

sage: cf30 = CyclotomicField(30)

sage: cf5 = CyclotomicField(5)

sage: cf3 = CyclotomicField(3)

sage: cf30.gen() + cf5.gen() + cf3.gen()

zeta30^6 + zeta30^5 + zeta30 - 1

sage: cf6 = CyclotomicField(6) ; z6 = cf6.0

sage: cf3 = CyclotomicField(3) ; z3 = cf3.0

sage: cf3(z6)

zeta3 + 1

sage: cf6(z3)

zeta6 - 1

sage: cf9 = CyclotomicField(9) ; z9 = cf9.0

sage: cf18 = CyclotomicField(18) ; z18 = cf18.0

sage: cf18(z9)

zeta18^2

sage: cf9(z18)

-zeta9^5

sage: cf18(z3)

zeta18^3 - 1

sage: cf18(z6)

zeta18^3

sage: cf18(z6)**2

zeta18^3 - 1

sage: cf9(z3)

zeta9^3

"""

def create_key(self, n=0, names=None, embedding=True):

r"""

Create the unique key for the cyclotomic field specified by the

parameters.

TESTS::

sage: CyclotomicField.create_key()

(0, None, True)

"""

n = ZZ(n)

if n < 0:

raise ValueError("n (=%s) must be a positive integer" % n)

if n > 0:

bracket = None

if embedding is True:

embedding = (CLF, (2 * CLF.pi() * CLF.gen() / n).exp())

elif embedding is not None:

x = number_field_morphisms.root_from_approx(QQ['x'].cyclotomic_polynomial(n), embedding)

embedding = (x.parent(), x)

if names is None:

names = "zeta%s"%n

names = normalize_names(1, names)

return n, names, embedding

def create_object(self, version, key, **extra_args):

r"""

Create the unique cyclotomic field defined by ``key``.

TESTS::

sage: CyclotomicField.create_object(None, (0, None, True))

Universal Cyclotomic Field

"""

n, names, embedding = key

if n == 0:

from sage.rings.universal_cyclotomic_field import UniversalCyclotomicField

return UniversalCyclotomicField()

else:

return NumberField_cyclotomic(n, names, embedding=embedding)

CyclotomicField = CyclotomicFieldFactory("sage.rings.number_field.number_field.CyclotomicField")

def is_CyclotomicField(x):

"""

Return True if x is a cyclotomic field, i.e., of the special

cyclotomic field class. This function does not return True for a

number field that just happens to be isomorphic to a cyclotomic

field.

EXAMPLES::

sage: from sage.rings.number_field.number_field import is_CyclotomicField

sage: is_CyclotomicField(NumberField(x^2 + 1,'zeta4'))

False

sage: is_CyclotomicField(CyclotomicField(4))

True

sage: is_CyclotomicField(CyclotomicField(1))

True

sage: is_CyclotomicField(QQ)

False

sage: is_CyclotomicField(7)

False

"""

return isinstance(x, NumberField_cyclotomic)

from . import number_field_base

is_NumberField = number_field_base.is_NumberField

class NumberField_generic(WithEqualityById, number_field_base.NumberField):

"""

Generic class for number fields defined by an irreducible

polynomial over `\\QQ`.

EXAMPLES::

sage: K.<a> = NumberField(x^3 - 2); K

Number Field in a with defining polynomial x^3 - 2

sage: TestSuite(K).run()

TESTS::

sage: k.<a> = NumberField(x^3 + 2); m. = NumberField(x^3 + 2)

sage: k == QQ

False

sage: k.<a> = NumberField(x^3 + 2); m.<a> = NumberField(x^3 + 2)

sage: k is m

True

sage: loads(dumps(k)) is k

True

sage: x = QQ['x'].gen()

sage: y = ZZ['y'].gen()

sage: K = NumberField(x^3 + x + 3, 'a'); K

Number Field in a with defining polynomial x^3 + x + 3

sage: K.defining_polynomial().parent()

Univariate Polynomial Ring in x over Rational Field

sage: L = NumberField(y^3 + y + 3, 'a'); L

Number Field in a with defining polynomial y^3 + y + 3

sage: L.defining_polynomial().parent()

Univariate Polynomial Ring in y over Rational Field

sage: L == K

False

sage: NumberField(ZZ['x'].0^4 + 23, 'a') == NumberField(ZZ['y'].0^4 + 23, 'a')

False

sage: NumberField(ZZ['x'].0^4 + 23, 'a') == NumberField(QQ['y'].0^4 + 23, 'a')

False

sage: NumberField(QQ['x'].0^4 + 23, 'a') == NumberField(QQ['y'].0^4 + 23, 'a')

False

sage: x = polygen(QQ); y = ZZ['y'].gen()

sage: NumberField(x^3 + x + 5, 'a') == NumberField(y^3 + y + 5, 'a')

False

sage: NumberField(x^3 + x + 5, 'a') == NumberField(y^4 + y + 5, 'a')

False

sage: NumberField(x^3 + x + 5, 'a') == NumberField(x^3 + x + 5, 'b')

False

sage: QuadraticField(2, 'a', embedding=2) == QuadraticField(2, 'a', embedding=-2)

False

sage: K.<a> = QuadraticField(2)

sage: R.<x> = K[]

sage: L. = K.extension(x^2+1)

sage: M. = L.absolute_field()

sage: M == L

False

sage: M['x'] == L['x']

False

sage: R.<x> = QQ[]

sage: R.<y> = QQ[]

sage: K.<a> = NumberField(x^2+1)

sage: L.<a> = NumberField(y^2+1)

sage: K == L

False

sage: hash(K) == hash(L)

False

Two relative number fields which are isomorphic as absolute

fields, but which are not presented the same way, are not

considered equal (see :trac:`18942`)::

sage: F.<omega> = NumberField(x^2 + x + 1)

sage: y = polygen(F)

sage: K = F.extension(y^3 + 3*omega + 2, 'alpha')

sage: L = F.extension(y^3 - 3*omega - 1, 'alpha')

sage: K == L

False

sage: K.is_isomorphic(L)

True

sage: hash(K) == hash(L)

False

This example illustrates the issue resolved in :trac:`18942`::

sage: F.<omega> = NumberField(x^2+x+1)

sage: xx = polygen(F)

sage: ps = [p for p, _ in F(7).factor()]

sage: for mu in ps:

....: K = F.extension(xx^3 - mu, 'alpha')

....: print(K.defining_polynomial().roots(K))

[(alpha, 1), ((-omega - 1)*alpha, 1), (omega*alpha, 1)]

[(alpha, 1), (omega*alpha, 1), ((-omega - 1)*alpha, 1)]

sage: for mu in ps:

....: K = F.extension(xx^3 - mu, 'alpha')

....: print(K.defining_polynomial().roots(K))

[(alpha, 1), ((-omega - 1)*alpha, 1), (omega*alpha, 1)]

[(alpha, 1), (omega*alpha, 1), ((-omega - 1)*alpha, 1)]

This example was suggested on sage-nt; see :trac:`18942`::

sage: G = DirichletGroup(80)

sage: for chi in G:

....: D = ModularSymbols(chi, 2, -1).cuspidal_subspace().new_subspace().decomposition()

....: for f in D:

....: elt = f.q_eigenform(10, 'alpha')[3]

....: assert elt.is_integral()

"""

def __init__(self, polynomial, name, latex_name,

check=True, embedding=None, category=None,

assume_disc_small=False, maximize_at_primes=None, structure=None):

"""

Create a number field.

EXAMPLES::

sage: NumberField(x^97 - 19, 'a')

Number Field in a with defining polynomial x^97 - 19

The defining polynomial must be irreducible::

sage: K.<a> = NumberField(x^2 - 1)

Traceback (most recent call last):

...

ValueError: defining polynomial (x^2 - 1) must be irreducible

If you use check=False, you avoid checking irreducibility of the

defining polynomial, which can save time.

sage: K.<a> = NumberField(x^2 - 1, check=False)

It can also be dangerous::

sage: (a-1)*(a+1)

The constructed object is in the category of number fields::

sage: NumberField(x^2 + 3, 'a').category()

Category of number fields

sage: category(NumberField(x^2 + 3, 'a'))

Category of number fields

The special types of number fields, e.g., quadratic fields, do

not have (yet?) their own category::

sage: QuadraticField(2,'d').category()

Category of number fields

TESTS::

sage: NumberField(ZZ['x'].0^4 + 23, 'a')

Number Field in a with defining polynomial x^4 + 23

sage: NumberField(QQ['x'].0^4 + 23, 'a')

Number Field in a with defining polynomial x^4 + 23

sage: NumberField(GF(7)['x'].0^4 + 23, 'a')

Traceback (most recent call last):

...

TypeError: polynomial must be defined over rational field

"""

self._assume_disc_small = assume_disc_small

self._maximize_at_primes = maximize_at_primes

self._structure = structure

default_category = _NumberFields

if category is None:

category = default_category

else:

assert category.is_subcategory(default_category), "%s is not a subcategory of %s"%(category, default_category)

ParentWithGens.__init__(self, QQ, name, category=category)

if not isinstance(polynomial, polynomial_element.Polynomial):

raise TypeError("polynomial (=%s) must be a polynomial"%repr(polynomial))

if check:

if not polynomial.parent().base_ring() == QQ:

raise TypeError("polynomial must be defined over rational field")

if not polynomial.is_irreducible():

raise ValueError("defining polynomial (%s) must be irreducible"%polynomial)

self._assign_names(name)

self.__latex_variable_name = latex_name

self.__polynomial = polynomial

self._pari_bnf_certified = False

self._integral_basis_dict = {}

if embedding is not None:

# Since Trac #20827, an embedding is specified as a pair

# (parent, x) with x the image of the distinguished

# generator (previously, it was just given as x). This

# allows the UniqueFactory to distinguish embeddings into

# different fields with images of the generator that

# compare equal.

# We allow both formats to support old pickles.

if isinstance(embedding, tuple):

parent, x = embedding

else:

parent, x = embedding.parent(), embedding

embedding = number_field_morphisms.NumberFieldEmbedding(self, parent, x)

self._populate_coercion_lists_(embedding=embedding, convert_method_name='_number_field_')

def _convert_map_from_(self, other):

r"""

Additional conversion maps from ``other`` may be defined by

:meth:`structure`.

.. SEEALSO::

:meth:`structure.NumberFieldStructure.create_structure`

TESTS::

sage: K. = QuadraticField(-1)

sage: L.<j> = K.change_names()

sage: L(i)

sage: K(j)

This also works for relative number fields and their absolute fields::

sage: K.<a> = QuadraticField(2)

sage: L. = K.extension(x^2 + 1)

sage: M. = L.absolute_field()

sage: M(i)

1/6*b^3 + 1/6*b

sage: L(b)

i - a

"""

from sage.categories.map import is_Map

if self._structure is not None:

structure = self.structure()

if len(structure) >= 2:

to_self = structure[1]

if is_Map(to_self) and to_self.domain() is other:

return to_self

if isinstance(other, NumberField_generic) and other._structure is not None:

structure = other.structure()

if len(structure) >= 1:

from_other = structure[0]

if is_Map(from_other) and from_other.codomain() is self:

return from_other

@cached_method

def _magma_polynomial_(self, magma):

"""

Return Magma version of the defining polynomial of this number field.

EXAMPLES::

sage: R.<x> = QQ[] # optional - magma

sage: K.<a> = NumberField(x^3+2) # optional - magma

sage: K._magma_polynomial_(magma) # optional - magma

x^3 + 2

sage: magma2=Magma() # optional - magma

sage: K._magma_polynomial_(magma2) # optional - magma

x^3 + 2

sage: K._magma_polynomial_(magma) is K._magma_polynomial_(magma) # optional - magma

True

sage: K._magma_polynomial_(magma) is K._magma_polynomial_(magma2) # optional - magma

False

"""

# NB f must not be garbage-collected, otherwise the

# return value of this function is invalid

return magma(self.defining_polynomial())

def _magma_init_(self, magma):

"""

Return a Magma version of this number field.

EXAMPLES::

sage: R.<t> = QQ[]

sage: K.<a> = NumberField(t^2 + 1)

sage: K._magma_init_(magma) # optional - magma

'SageCreateWithNames(NumberField(_sage_[...]),["a"])'

sage: L = magma(K) # optional - magma

sage: L # optional - magma

Number Field with defining polynomial t^2 + 1 over the Rational Field

sage: L.sage() # optional - magma

Number Field in a with defining polynomial t^2 + 1

sage: L.sage() is K # optional - magma

True

sage: L.1 # optional - magma

sage: L.1^2 # optional - magma

-1

sage: m = magma(a+1/2); m # optional - magma

1/2*(2*a + 1)

sage: m.sage() # optional - magma

a + 1/2

A relative number field::

sage: S. = K[]

sage: M. = NumberField(u^3+u+a)

sage: L = magma(M) # optional - magma

sage: L # optional - magma

Number Field with defining polynomial u^3 + u + a over its ground field

sage: L.sage() is M # optional - magma

True

"""

# Get magma version of defining polynomial of this number field

f = self._magma_polynomial_(magma)

s = 'NumberField(%s)'%f.name()

return magma._with_names(s, self.variable_names())

def construction(self):

r"""

Construction of self

EXAMPLES::

sage: K.<a>=NumberField(x^3+x^2+1,embedding=CC.gen())

sage: F,R = K.construction()

sage: F

AlgebraicExtensionFunctor

sage: R

Rational Field

The construction functor respects distinguished embeddings::

sage: F(R) is K

True

sage: F.embeddings

[0.2327856159383841? + 0.7925519925154479?*I]

TESTS::

sage: K.<a> = NumberField(x^3+x+1)

sage: R.<t> = ZZ[]

sage: a+t # indirect doctest

t + a

sage: (a+t).parent()

Univariate Polynomial Ring in t over Number Field in a with defining polynomial x^3 + x + 1

The construction works for non-absolute number fields as well::

sage: K.<a,b,c>=NumberField([x^3+x^2+1,x^2+1,x^7+x+1])

sage: F,R = K.construction()

sage: F(R) == K

True

sage: P.<x> = QQ[]

sage: K.<a> = NumberField(x^3-5,embedding=0)

sage: L. = K.extension(x^2+a)

sage: a*b

a*b

"""

from sage.categories.pushout import AlgebraicExtensionFunctor

from sage.all import QQ

names = self.variable_names()

polys = []

embeddings = []

structures = []

K = self

while K is not QQ:

polys.append(K.relative_polynomial())

embeddings.append(None if K.coerce_embedding() is None else K.coerce_embedding()(K.gen()))

structures.append(K._structure)

K = K.base_field()

return (AlgebraicExtensionFunctor(polys, names, embeddings, structures), QQ)

def _element_constructor_(self, x, check=True):

r"""

Convert ``x`` into an element of this number field.

INPUT:

- ``x`` -- Sage (or Python) object

OUTPUT:

A :class:`~number_field_element.NumberFieldElement`

constructed from ``x``.

TESTS::

sage: K.<a> = NumberField(x^3 + 17)

sage: K(a) is a # indirect doctest

True

sage: K('a^2 + 2/3*a + 5')

a^2 + 2/3*a + 5

sage: K('1').parent()

Number Field in a with defining polynomial x^3 + 17

sage: K(3/5).parent()

Number Field in a with defining polynomial x^3 + 17

sage: K.<a> = NumberField(polygen(QQ)^2 - 5)

sage: F. = K.extension(polygen(K))

sage: F([a])

We can create number field elements from PARI::

sage: K.<a> = NumberField(x^3 - 17)

sage: K(pari(42))

sage: K(pari("5/3"))

5/3

sage: K(pari("[3/2, -5, 0]~")) # Uses Z-basis

-5/3*a^2 + 5/3*a - 1/6

From a PARI polynomial or ``POLMOD``, note that the variable

name does not matter::

sage: K(pari("-5/3*q^2 + 5/3*q - 1/6"))

-5/3*a^2 + 5/3*a - 1/6

sage: K(pari("Mod(-5/3*q^2 + 5/3*q - 1/6, q^3 - 17)"))

-5/3*a^2 + 5/3*a - 1/6

sage: K(pari("x^5/17"))

a^2

An error is raised when a PARI element with an incorrect

modulus is given:

sage: K(pari("Mod(-5/3*q^2 + 5/3*q - 1/6, q^3 - 999)"))

Traceback (most recent call last):

...

TypeError: cannot convert PARI element Mod(-5/3*q^2 + 5/3*q - 1/6, q^3 - 999) into Number Field in a with defining polynomial x^3 - 17

Test round-trip conversion to PARI and back::

sage: x = polygen(QQ)

sage: K.<a> = NumberField(x^3 - 1/2*x + 1/3)

sage: b = K.random_element()

sage: K(pari(b)) == b

True

sage: F.<c> = NumberField(2*x^3 + x + 1)

sage: d = F.random_element()

sage: F(F.pari_nf().nfalgtobasis(d)) == d

True

If the PARI polynomial is different from the Sage polynomial,

a warning is printed unless ``check=False`` is specified::

sage: b = pari(a); b

Mod(-1/12*y^2 - 1/12*y + 1/6, y^3 - 3*y - 22)

sage: K(b.lift())

doctest:...: UserWarning: interpreting PARI polynomial -1/12*y^2 - 1/12*y + 1/6 relative to the defining polynomial x^3 - 3*x - 22 of the PARI number field

sage: K(b.lift(), check=False)

Using a GAP element may be tricky, as it may contain

an exclamation mark::

sage: L.<tau> = NumberField(x^3-2)

sage: gap(tau^3)

sage: gap(tau)^3

sage: L(gap(tau)^3) # indirect doctest

Check that :trac:`22202` is fixed::

sage: y = QQ['y'].gen()

sage: R = QQ.extension(y^2-2,'a')['x']

sage: R("a*x").factor()

(a) * x

"""

if isinstance(x, number_field_element.NumberFieldElement):

K = x.parent()

if K is self:

return x

elif isinstance(x, (number_field_element.OrderElement_absolute,

number_field_element.OrderElement_relative,

number_field_element_quadratic.OrderElement_quadratic)):

L = K.number_field()

if L is self:

return self._element_class(self, x)

x = L(x)

return self._coerce_from_other_number_field(x)

elif isinstance(x, pari_gen):

if x.type() == "t_POLMOD":

modulus = x.mod()

if check and modulus != self.pari_polynomial(modulus.variable()):

raise TypeError("cannot convert PARI element %s into %s" % (x, self))

x = x.lift()

check = False

elif x.type() == "t_COL":

x = self.pari_nf().nfbasistoalg_lift(x)

check = False

if x.type() in ["t_INT", "t_FRAC"]:

pass

elif x.type() == "t_POL":

var = self.absolute_polynomial().variable_name()

if check and self.pari_polynomial(var) != self.absolute_polynomial().monic():

from warnings import warn

warn("interpreting PARI polynomial %s relative to the defining polynomial %s of the PARI number field"

% (x, self.pari_polynomial()))

# We consider x as a polynomial in the standard

# generator of the PARI number field, and convert it

# to a polynomial in the Sage generator.

if x.poldegree() > 0:

beta = self._pari_absolute_structure()[2]

x = x(beta).lift()

else:

raise TypeError("%s has unsupported PARI type %s" % (x, x.type()))

x = self.absolute_polynomial().parent()(x)

return self._element_class(self, x)

elif sage.interfaces.gap.is_GapElement(x):

s = x._sage_repr()

if self.variable_name() in s:

return self._convert_from_str(s)

return self._convert_from_str(s.replace('!', ''))

elif isinstance(x,str):

return self._convert_from_str(x)

elif isinstance(x, (tuple, list)) or \

(isinstance(x, sage.modules.free_module_element.FreeModuleElement) and

self.base_ring().has_coerce_map_from(x.parent().base_ring())):

if len(x) != self.relative_degree():

raise ValueError("Length must be equal to the degree of this number field")

result = x[0]

for i in range(1, self.relative_degree()):

result += x[i]*self.gen(0)**i

return result

return self._convert_non_number_field_element(x)

def _convert_non_number_field_element(self, x):

"""

Convert a non-number field element ``x`` into this number field.

INPUT:

- ``x`` -- a non number field element, e.g., a list, integer,

rational, or polynomial

EXAMPLES::

sage: K.<a> = NumberField(x^3 + 2/3)

sage: K._convert_non_number_field_element(-7/8)

-7/8

sage: K._convert_non_number_field_element([1,2,3])

3*a^2 + 2*a + 1

The list is just turned into a polynomial in the generator::

sage: K._convert_non_number_field_element([0,0,0,1,1])

-2/3*a - 2/3

Any polynomial whose coefficients can be converted to rationals

will convert to the number field, e.g., this one in

characteristic 7::

sage: f = GF(7)['y']([1,2,3]); f

3*y^2 + 2*y + 1

sage: K._convert_non_number_field_element(f)

3*a^2 + 2*a + 1

But not this one over a field of order 27::

sage: F27.<g> = GF(27)

sage: f = F27['z']([g^2, 2*g, 1]); f

z^2 + 2*g*z + g^2

sage: K._convert_non_number_field_element(f)

Traceback (most recent call last):

...

TypeError: unable to convert g^2 to a rational

One can also convert an element of the polynomial quotient ring

that is isomorphic to the number field::

sage: K.<a> = NumberField(x^3 + 17)

sage: b = K.polynomial_quotient_ring().random_element()

sage: K(b)

-1/2*a^2 - 4

We can convert symbolic expressions::

sage: I = sqrt(-1); parent(I)

Symbolic Ring

sage: GaussianIntegers()(2 + I)

I + 2

sage: K1 = QuadraticField(3)

sage: K2 = QuadraticField(5)

sage: (K,) = K1.composite_fields(K2, preserve_embedding=True)

sage: K(sqrt(3) + sqrt(5))

-1/2*a0^3 + 8*a0

sage: K(sqrt(-3)*I)

1/4*a0^3 - 7/2*a0

"""

if isinstance(x, integer_types + (Rational, Integer, pari_gen, list)):

return self._element_class(self, x)

if isinstance(x, sage.rings.polynomial.polynomial_quotient_ring_element.PolynomialQuotientRingElement)\

and (x in self.polynomial_quotient_ring()):

y = self.polynomial_ring().gen()

return x.lift().subs({y:self.gen()})

if isinstance(x, (sage.rings.qqbar.AlgebraicNumber, sage.rings.qqbar.AlgebraicReal)):

return self._convert_from_qqbar(x)

if isinstance(x, polynomial_element.Polynomial):

return self._element_class(self, x)

# Try converting via QQ.

try:

y = QQ(x)

except (TypeError, ValueError):

pass

else:

return self._element_class(self, y)

# Final attempt: convert via QQbar. This deals in particular

# with symbolic expressions like sqrt(-5).

try:

y = sage.rings.qqbar.QQbar(x)

except (TypeError, ValueError):

pass

else:

return self._convert_from_qqbar(y)

raise TypeError("unable to convert %r to %s" % (x, self))

def _convert_from_qqbar(self, x):

"""

Convert an element of ``QQbar`` or ``AA`` to this number field,

if possible.

This requires that the given number field is equipped with an

embedding.

INPUT:

- ``x`` -- an algebraic number

EXAMPLES::

sage: K.<a> = QuadraticField(3)

sage: K._convert_from_qqbar(7 + 2*AA(3).sqrt())

2*a + 7

sage: GaussianIntegers()(QQbar(I))

sage: CyclotomicField(15)(QQbar.zeta(5))

zeta15^3

sage: CyclotomicField(12)(QQbar.zeta(5))

Traceback (most recent call last):

...

TypeError: unable to convert 0.3090169943749474? + 0.9510565162951536?*I to Cyclotomic Field of order 12 and degree 4

"""

# We use the diagram

# self

# ↑ ↘

# F → QQbar

# Where F is the smallest number field containing x.

# y is the pre-image such that x = F(y)

F, y, F_to_QQbar = x.as_number_field_element(minimal=True)

# Try all embeddings from F into self

from sage.rings.qqbar import QQbar

for F_to_self in F.embeddings(self):

z = F_to_self(y)

# Check whether the diagram commutes

if QQbar(z) == x:

return z

raise TypeError("unable to convert %r to %s" % (x, self))

def _convert_from_str(self, x):

"""

Coerce a string representation of an element of this

number field into this number field.

INPUT:

x -- string

EXAMPLES::

sage: k.<theta25> = NumberField(x^3+(2/3)*x+1)

sage: k._convert_from_str('theta25^3 + (1/3)*theta25')

-1/3*theta25 - 1

This function is called by the coerce method when it gets a string

as input:

sage: k('theta25^3 + (1/3)*theta25')

-1/3*theta25 - 1

"""

w = sage.misc.all.sage_eval(x,locals=self.gens_dict())

if not (is_Element(w) and w.parent() is self):

return self(w)

else:

return w

def _Hom_(self, codomain, category=None):

"""

Return homset of homomorphisms from self to the number field codomain.

EXAMPLES:

This method is implicitly called by :meth:`Hom` and

:meth:`sage.categories.homset.Hom`::

sage: K. = NumberField(x^2 + 1); K

Number Field in i with defining polynomial x^2 + 1

sage: K.Hom(K) # indirect doctest

Automorphism group of Number Field in i with defining polynomial x^2 + 1

sage: Hom(K, QuadraticField(-1, 'b'))

Set of field embeddings from Number Field in i with defining polynomial x^2 + 1 to Number Field in b with defining polynomial x^2 + 1

CHECKME: handling of the case where codomain is not a number field?

sage: Hom(K, VectorSpace(QQ,3))

Set of Morphisms from Number Field in i with defining polynomial x^2 + 1 to Vector space of dimension 3 over Rational Field in Category of commutative additive groups

TESTS:

Verify that :trac:`22001` has been resolved::

sage: R.<x> = QQ[]

sage: K.<a> = QQ.extension(x^2 + 1)

sage: K.hom([a]).category_for()

Category of number fields

sage: H = End(K)

sage: loads(dumps(H)) is H

True

"""

if not is_NumberFieldHomsetCodomain(codomain):

raise TypeError("{} is not suitable as codomain for homomorphisms from {}".format(codomain, self))

from .morphism import NumberFieldHomset

return NumberFieldHomset(self, codomain, category)

@cached_method

def structure(self):

"""

Return fixed isomorphism or embedding structure on self.

This is used to record various isomorphisms or embeddings that

arise naturally in other constructions.

EXAMPLES::

sage: K.<z> = NumberField(x^2 + 3)

sage: L.<a> = K.absolute_field(); L

Number Field in a with defining polynomial x^2 + 3

sage: L.structure()

(Isomorphism given by variable name change map:

From: Number Field in a with defining polynomial x^2 + 3

To: Number Field in z with defining polynomial x^2 + 3,

Isomorphism given by variable name change map:

From: Number Field in z with defining polynomial x^2 + 3

To: Number Field in a with defining polynomial x^2 + 3)

sage: K.<a> = QuadraticField(-3)

sage: R.<y> = K[]

sage: D.<x0> = K.extension(y)

sage: D_abs.<y0> = D.absolute_field()

sage: D_abs.structure()[0](y0)

-a

"""

if self._structure is None:

f = self.hom(self)

return f,f

else:

return self._structure.create_structure(self)

def completion(self, p, prec, extras={}):

"""

Returns the completion of self at `p` to the specified

precision. Only implemented at archimedean places, and then only if

an embedding has been fixed.

EXAMPLES::

sage: K.<a> = QuadraticField(2)

sage: K.completion(infinity, 100)

Real Field with 100 bits of precision

sage: K.<zeta> = CyclotomicField(12)

sage: K.completion(infinity, 53, extras={'type': 'RDF'})

Complex Double Field

sage: zeta + 1.5 # implicit test

2.36602540378444 + 0.500000000000000*I

"""

if p == infinity.infinity:

gen_image = self.gen_embedding()

if gen_image is not None:

if gen_image in RDF:

return QQ.completion(p, prec, extras)

elif gen_image in CDF:

return QQ.completion(p, prec, extras).algebraic_closure()

raise ValueError("No embedding into the complex numbers has been specified.")

else:

raise NotImplementedError

def primitive_element(self):

r"""

Return a primitive element for this field, i.e., an element that

generates it over `\QQ`.

EXAMPLES::

sage: K.<a> = NumberField(x^3 + 2)

sage: K.primitive_element()

sage: K.<a,b,c> = NumberField([x^2-2,x^2-3,x^2-5])

sage: K.primitive_element()

a - b + c

sage: alpha = K.primitive_element(); alpha

a - b + c

sage: alpha.minpoly()

x^2 + (2*b - 2*c)*x - 2*c*b + 6

sage: alpha.absolute_minpoly()

x^8 - 40*x^6 + 352*x^4 - 960*x^2 + 576

"""

try:

return self.__primitive_element

except AttributeError:

pass

K = self.absolute_field('a')

from_K, to_K = K.structure()

self.__primitive_element = from_K(K.gen())

return self.__primitive_element

def random_element(self, num_bound=None, den_bound=None,

integral_coefficients=False, distribution=None):

r"""

Return a random element of this number field.

INPUT:

- ``num_bound`` - Bound on numerator of the coefficients of

the resulting element

- ``den_bound`` - Bound on denominators of the coefficients

of the resulting element

- ``integral_coefficients`` (default: False) - If True, then

the resulting element will have integral

coefficients. This option overrides any

value of `den_bound`.

- ``distribution`` - Distribution to use for the coefficients

of the resulting element

OUTPUT:

- Element of this number field

EXAMPLES::

sage: K.<j> = NumberField(x^8+1)

sage: K.random_element()

1/2*j^7 - j^6 - 12*j^5 + 1/2*j^4 - 1/95*j^3 - 1/2*j^2 - 4

sage: K.<a,b,c> = NumberField([x^2-2,x^2-3,x^2-5])

sage: K.random_element()

((6136*c - 7489/3)*b + 5825/3*c - 71422/3)*a + (-4849/3*c + 58918/3)*b - 45718/3*c + 75409/12

sage: K.<a> = NumberField(x^5-2)

sage: K.random_element(integral_coefficients=True)

a^3 + a^2 - 3*a - 1

TESTS::

sage: K.<a> = NumberField(x^5-2)

sage: K.random_element(-1)

Traceback (most recent call last):

...

TypeError: x must be < y

sage: K.random_element(5,0)

Traceback (most recent call last):

...

TypeError: x must be < y

sage: QQ[I].random_element(0)

Traceback (most recent call last):

...

TypeError: x must be > 0

"""

if integral_coefficients:

den_bound = 1

return self._zero_element._random_element(num_bound=num_bound,

den_bound=den_bound,

distribution=distribution)

def subfield(self, alpha, name=None, names=None):

r"""

Return a number field `K` isomorphic to `\QQ(\alpha)`

(if this is an absolute number field) or `L(\alpha)` (if this

is a relative extension `M/L`) and a map from K to self that

sends the generator of K to alpha.

INPUT:

- ``alpha`` - an element of self, or something that

coerces to an element of self.

OUTPUT:

- ``K`` - a number field

- ``from_K`` - a homomorphism from K to self that

sends the generator of K to alpha.

EXAMPLES::

sage: K.<a> = NumberField(x^4 - 3); K

Number Field in a with defining polynomial x^4 - 3

sage: H., from_H = K.subfield(a^2)

sage: H

Number Field in b with defining polynomial x^2 - 3

sage: from_H(b)

a^2

sage: from_H

Ring morphism:

From: Number Field in b with defining polynomial x^2 - 3

To: Number Field in a with defining polynomial x^4 - 3

Defn: b |--> a^2

A relative example. Note that the result returned is the subfield generated

by `\alpha` over ``self.base_field()``, not over `\QQ` (see :trac:`5392`)::

sage: L.<a> = NumberField(x^2 - 3)

sage: M. = L.extension(x^4 + 1)

sage: K, phi = M.subfield(b^2)

sage: K.base_field() is L

True

Subfields inherit embeddings::

sage: K.<z> = CyclotomicField(5)

sage: L, K_from_L = K.subfield(z-z^2-z^3+z^4)

sage: L

Number Field in z0 with defining polynomial x^2 - 5

sage: CLF_from_K = K.coerce_embedding(); CLF_from_K

Generic morphism:

From: Cyclotomic Field of order 5 and degree 4

To: Complex Lazy Field

Defn: z -> 0.309016994374948? + 0.951056516295154?*I

sage: CLF_from_L = L.coerce_embedding(); CLF_from_L

Generic morphism:

From: Number Field in z0 with defining polynomial x^2 - 5

To: Complex Lazy Field

Defn: z0 -> 2.236067977499790?

Check transitivity::

sage: CLF_from_L(L.gen())

2.236067977499790?

sage: CLF_from_K(K_from_L(L.gen()))

2.23606797749979? + 0.?e-14*I

If `self` has no specified embedding, then `K` comes with an

embedding in `self`::

sage: K.<a> = NumberField(x^6 - 6*x^4 + 8*x^2 - 1)

sage: L., from_L = K.subfield(a^2)

sage: L

Number Field in b with defining polynomial x^3 - 6*x^2 + 8*x - 1

sage: L.gen_embedding()

a^2

You can also view a number field as having a different generator by

just choosing the input to generate the whole field; for that it is

better to use ``self.change_generator``, which gives

isomorphisms in both directions.

"""

if not names is None:

name = names

if name is None:

name = self.variable_name() + '0'

beta = self(alpha)

f = beta.minpoly()

# If self has a specified embedding, K should inherit it

if self.coerce_embedding() is not None:

emb = self.coerce_embedding()(beta)

else:

# Otherwise K should at least come with an embedding in self

emb = beta

K = NumberField(f, names=name, embedding=emb)

from_K = K.hom([beta])

return K, from_K

def change_generator(self, alpha, name=None, names=None):

r"""

Given the number field self, construct another isomorphic number

field `K` generated by the element alpha of self, along

with isomorphisms from `K` to self and from self to

`K`.

EXAMPLES::

sage: L. = NumberField(x^2 + 1); L

Number Field in i with defining polynomial x^2 + 1

sage: K, from_K, to_K = L.change_generator(i/2 + 3)

sage: K

Number Field in i0 with defining polynomial x^2 - 6*x + 37/4

sage: from_K

Ring morphism:

From: Number Field in i0 with defining polynomial x^2 - 6*x + 37/4

To: Number Field in i with defining polynomial x^2 + 1

Defn: i0 |--> 1/2*i + 3

sage: to_K

Ring morphism:

From: Number Field in i with defining polynomial x^2 + 1

To: Number Field in i0 with defining polynomial x^2 - 6*x + 37/4

Defn: i |--> 2*i0 - 6

We can also do

sage: K.<c>, from_K, to_K = L.change_generator(i/2 + 3); K

Number Field in c with defining polynomial x^2 - 6*x + 37/4

We compute the image of the generator `\sqrt{-1}` of `L`.

sage: to_K(i)

2*c - 6

Note that the image is indeed a square root of -1.

sage: to_K(i)^2

-1

sage: from_K(to_K(i))

sage: to_K(from_K(c))

"""

if not names is None:

name = names

alpha = self(alpha)

K, from_K = self.subfield(alpha, name=name)

if K.degree() != self.degree():

raise ValueError("alpha must generate a field of degree %s, but alpha generates a subfield of degree %s"%(self.degree(), K.degree()))

# Now compute to_K, which is an isomorphism

# from self to K such that from_K(to_K(x)) == x for all x,

# and to_K(from_K(y)) == y.

# To do this, we must compute the image of self.gen()

# under to_K. This means writing self.gen() as a

# polynomial in alpha, which is possible by the degree

# check above. This latter we do by linear algebra.

phi = alpha.coordinates_in_terms_of_powers()

c = phi(self.gen())

to_K = self.hom([K(c)])

return K, from_K, to_K

def is_absolute(self):

"""

Returns True if self is an absolute field.

This function will be implemented in the derived classes.

EXAMPLES::

sage: K = CyclotomicField(5)

sage: K.is_absolute()

True

"""

raise NotImplementedError

def is_relative(self):

"""

EXAMPLES::

sage: K.<a> = NumberField(x^10 - 2)

sage: K.is_absolute()

True

sage: K.is_relative()

False

"""

return not self.is_absolute()

@cached_method

def absolute_field(self, names):

"""

Returns self as an absolute extension over QQ.

OUTPUT:

- ``K`` - this number field (since it is already

absolute)

Also, ``K.structure()`` returns from_K and to_K,

where from_K is an isomorphism from K to self and to_K is an

isomorphism from self to K.

EXAMPLES::

sage: K = CyclotomicField(5)

sage: K.absolute_field('a')

Number Field in a with defining polynomial x^4 + x^3 + x^2 + x + 1

"""

return NumberField(self.defining_polynomial(), names, check=False, structure=structure.NameChange(self))

def is_isomorphic(self, other, isomorphism_maps = False):

"""

Return True if self is isomorphic as a number field to other.

EXAMPLES::

sage: k.<a> = NumberField(x^2 + 1)

sage: m. = NumberField(x^2 + 4)

sage: k.is_isomorphic(m)

True

sage: m. = NumberField(x^2 + 5)

sage: k.is_isomorphic (m)

False

sage: k = NumberField(x^3 + 2, 'a')

sage: k.is_isomorphic(NumberField((x+1/3)^3 + 2, 'b'))

True

sage: k.is_isomorphic(NumberField(x^3 + 4, 'b'))

True

sage: k.is_isomorphic(NumberField(x^3 + 5, 'b'))

False

sage: k = NumberField(x^2 - x - 1, 'b')

sage: l = NumberField(x^2 - 7, 'a')

sage: k.is_isomorphic(l, True)

(False, [])

sage: k = NumberField(x^2 - x - 1, 'b')

sage: ky.<y> = k[];

sage: l = NumberField(y, 'a')

sage: k.is_isomorphic(l, True)

(True, [-x, x + 1])

"""

if not isinstance(other, NumberField_generic):

raise ValueError("other must be a generic number field.")

t = self.pari_polynomial().nfisisom(other.pari_polynomial())

if t == 0:

t = []

res = False

else:

res = True

if isomorphism_maps:

return res, t

else:

return res

def is_totally_real(self):

"""

Return True if self is totally real, and False otherwise.

Totally real means that every isomorphic embedding of self into the

complex numbers has image contained in the real numbers.

EXAMPLES::

sage: NumberField(x^2+2, 'alpha').is_totally_real()

False

sage: NumberField(x^2-2, 'alpha').is_totally_real()

True

sage: NumberField(x^4-2, 'alpha').is_totally_real()

False

"""

return self.signature()[1] == 0

def is_totally_imaginary(self):

"""

Return True if self is totally imaginary, and False otherwise.

Totally imaginary means that no isomorphic embedding of self into

the complex numbers has image contained in the real numbers.

EXAMPLES::

sage: NumberField(x^2+2, 'alpha').is_totally_imaginary()

True

sage: NumberField(x^2-2, 'alpha').is_totally_imaginary()

False

sage: NumberField(x^4-2, 'alpha').is_totally_imaginary()

False

"""

return self.signature()[0] == 0

def is_CM(self):

r"""

Return True if self is a CM field (i.e. a totally imaginary

quadratic extension of a totally real field).

EXAMPLES::

sage: Q.<a> = NumberField(x - 1)

sage: Q.is_CM()

False

sage: K. = NumberField(x^2 + 1)

sage: K.is_CM()

True

sage: L.<zeta20> = CyclotomicField(20)

sage: L.is_CM()

True

sage: K.<omega> = QuadraticField(-3)

sage: K.is_CM()

True

sage: L.<sqrt5> = QuadraticField(5)

sage: L.is_CM()

False

sage: F.<a> = NumberField(x^3 - 2)

sage: F.is_CM()

False

sage: F.<a> = NumberField(x^4-x^3-3*x^2+x+1)

sage: F.is_CM()

False

The following are non-CM totally imaginary fields.

sage: F.<a> = NumberField(x^4 + x^3 - x^2 - x + 1)

sage: F.is_totally_imaginary()

True

sage: F.is_CM()

False

sage: F2.<a> = NumberField(x^12 - 5*x^11 + 8*x^10 - 5*x^9 - \

x^8 + 9*x^7 + 7*x^6 - 3*x^5 + 5*x^4 + \

7*x^3 - 4*x^2 - 7*x + 7)

sage: F2.is_totally_imaginary()

True

sage: F2.is_CM()

False

The following is a non-cyclotomic CM field.

sage: M.<a> = NumberField(x^4 - x^3 - x^2 - 2*x + 4)

sage: M.is_CM()

True

Now, we construct a totally imaginary quadratic extension of a

totally real field (which is not cyclotomic).

sage: E_0.<a> = NumberField(x^7 - 4*x^6 - 4*x^5 + 10*x^4 + 4*x^3 - \

6*x^2 - x + 1)

sage: E_0.is_totally_real()

True

sage: E. = E_0.extension(x^2 + 1)

sage: E.is_CM()

True

Finally, a CM field that is given as an extension that is not CM.

sage: E_0.<a> = NumberField(x^2 - 4*x + 16)

sage: y = polygen(E_0)

sage: E.<z> = E_0.extension(y^2 - E_0.gen() / 2)

sage: E.is_CM()

True

sage: E.is_CM_extension()

False

"""

#Return cached answer if available

try:

return self.__is_CM

except(AttributeError):

pass

#Then, deal with simple cases

if is_odd(self.absolute_degree()):

self.__is_CM = False

return False

if isinstance(

self, sage.rings.number_field.number_field.NumberField_quadratic):

self.__is_CM = (self.discriminant() < 0)

return self.__is_CM

if isinstance(

self, sage.rings.number_field.number_field.NumberField_cyclotomic):

self.__is_CM = True

return True

if not self.is_totally_imaginary():

self.__is_CM = False

return False

if self.is_absolute():

K = self

else:

F = self.base_field()

if F.absolute_degree() == self.absolute_degree() / 2:

if F.is_totally_real():

self.__is_CM = True

self.__max_tot_real_sub = [F, self.coerce_map_from(F)]

return True

K = self.absolute_field('z')

#Check for index 2 subextensions that are totally real

possibilities = K.subfields(K.absolute_degree()/2)

for F, phi, _ in possibilities:

if F.is_totally_real():

self.__is_CM = True

if self.is_relative():

phi = phi.post_compose(K.structure()[0])

self.__max_tot_real_sub = [F, phi]

return True

self.__is_CM = False

return False

def complex_conjugation(self):

"""

Return the complex conjugation of self.

This is only well-defined for fields contained in CM fields

(i.e. for totally real fields and CM fields). Recall that a CM

field is a totally imaginary quadratic extension of a totally

real field. For other fields, a ValueError is raised.

EXAMPLES::

sage: QuadraticField(-1, 'I').complex_conjugation()

Ring endomorphism of Number Field in I with defining polynomial x^2 + 1

Defn: I |--> -I

sage: CyclotomicField(8).complex_conjugation()

Ring endomorphism of Cyclotomic Field of order 8 and degree 4

Defn: zeta8 |--> -zeta8^3

sage: QuadraticField(5, 'a').complex_conjugation()

Identity endomorphism of Number Field in a with defining polynomial x^2 - 5

sage: F = NumberField(x^4 + x^3 - 3*x^2 - x + 1, 'a')

sage: F.is_totally_real()

True

sage: F.complex_conjugation()

Identity endomorphism of Number Field in a with defining polynomial x^4 + x^3 - 3*x^2 - x + 1

sage: F. = NumberField(x^2 - 2)

sage: F.extension(x^2 + 1, 'a').complex_conjugation()

Relative number field endomorphism of Number Field in a with defining polynomial x^2 + 1 over its base field

Defn: a |--> -a

b |--> b

sage: F2. = NumberField(x^2 + 2)

sage: K2.<a> = F2.extension(x^2 + 1)

sage: cc = K2.complex_conjugation()

sage: cc(a)

-a

sage: cc(b)

-b

"""

#Return cached answer if available

try:

return self.__complex_conjugation

except(AttributeError):

pass

#Then, deal with simple cases

if isinstance(

self, sage.rings.number_field.number_field.NumberField_quadratic):

disc = self.discriminant()

if disc > 0:

self.__complex_conjugation = self.coerce_map_from(self)

return self.__complex_conjugation

else:

a = self.gen()

r = a.trace()

iy = a - r / 2

self.__complex_conjugation = self.hom([a - 2 * iy], check=False)

return self.__complex_conjugation

if isinstance(

self, sage.rings.number_field.number_field.NumberField_cyclotomic):

zeta = self.gen()

self.__complex_conjugation = self.hom([zeta ** (-1)], check=False)

return self.__complex_conjugation

if self.is_totally_real():

self.__complex_conjugation = self.coerce_map_from(self)

return self.__complex_conjugation

if not self.is_CM():

raise ValueError('Complex conjugation is only well-defined for fields contained in CM fields.')

#In the remaining case, self.is_CM() should have cached __max_tot_real_sub

try:

F, phi = self.__max_tot_real_sub

except(AttributeError):

F, phi = self.maximal_totally_real_subfield()

if self.is_absolute():

K_rel = self.relativize(phi, self.variable_name() * 2)

to_abs, from_abs = K_rel.structure()

self.__complex_conjugation = K_rel.automorphisms()[1].pre_compose( \

from_abs).post_compose(to_abs)

self.__complex_conjugation = self.hom([self.__complex_conjugation(self.gen())], check=False)

return self.__complex_conjugation

else:

if self.is_CM_extension():

return self.automorphisms()[1]

K_abs = self.absolute_field(self.variable_name() * 2)

to_self, from_self = K_abs.structure()

K_rel = K_abs.relativize(phi.post_compose(from_self), self.variable_name() * 3)

to_abs, from_abs = K_rel.structure()

self.__complex_conjugation = K_rel.automorphisms()[1].pre_compose(from_abs).post_compose(to_abs)

self.__complex_conjugation = K_abs.hom([self.__complex_conjugation(K_abs.gen())], check=False)

self.__complex_conjugation = self.__complex_conjugation.pre_compose(from_self).post_compose(to_self)

return self.__complex_conjugation

def maximal_totally_real_subfield(self):

"""

Return the maximal totally real subfield of self together with an embedding of it into self.

EXAMPLES::

sage: F.<a> = QuadraticField(11)

sage: F.maximal_totally_real_subfield()

[Number Field in a with defining polynomial x^2 - 11, Identity endomorphism of Number Field in a with defining polynomial x^2 - 11]

sage: F.<a> = QuadraticField(-15)

sage: F.maximal_totally_real_subfield()

[Rational Field, Natural morphism:

From: Rational Field

To: Number Field in a with defining polynomial x^2 + 15]

sage: F.<a> = CyclotomicField(29)

sage: F.maximal_totally_real_subfield()

(Number Field in a0 with defining polynomial x^14 + x^13 - 13*x^12 - 12*x^11 + 66*x^10 + 55*x^9 - 165*x^8 - 120*x^7 + 210*x^6 + 126*x^5 - 126*x^4 - 56*x^3 + 28*x^2 + 7*x - 1, Ring morphism:

From: Number Field in a0 with defining polynomial x^14 + x^13 - 13*x^12 - 12*x^11 + 66*x^10 + 55*x^9 - 165*x^8 - 120*x^7 + 210*x^6 + 126*x^5 - 126*x^4 - 56*x^3 + 28*x^2 + 7*x - 1

To: Cyclotomic Field of order 29 and degree 28

Defn: a0 |--> -a^27 - a^26 - a^25 - a^24 - a^23 - a^22 - a^21 - a^20 - a^19 - a^18 - a^17 - a^16 - a^15 - a^14 - a^13 - a^12 - a^11 - a^10 - a^9 - a^8 - a^7 - a^6 - a^5 - a^4 - a^3 - a^2 - 1)

sage: F.<a> = NumberField(x^3 - 2)

sage: F.maximal_totally_real_subfield()

[Rational Field, Coercion map:

From: Rational Field

To: Number Field in a with defining polynomial x^3 - 2]

sage: F.<a> = NumberField(x^4 - x^3 - x^2 + x + 1)

sage: F.maximal_totally_real_subfield()

[Rational Field, Coercion map:

From: Rational Field

To: Number Field in a with defining polynomial x^4 - x^3 - x^2 + x + 1]

sage: F.<a> = NumberField(x^4 - x^3 + 2*x^2 + x + 1)

sage: F.maximal_totally_real_subfield()

[Number Field in a1 with defining polynomial x^2 - x - 1, Ring morphism:

From: Number Field in a1 with defining polynomial x^2 - x - 1

To: Number Field in a with defining polynomial x^4 - x^3 + 2*x^2 + x + 1

Defn: a1 |--> -1/2*a^3 - 1/2]

sage: F.<a> = NumberField(x^4-4*x^2-x+1)

sage: F.maximal_totally_real_subfield()

[Number Field in a with defining polynomial x^4 - 4*x^2 - x + 1, Identity endomorphism of Number Field in a with defining polynomial x^4 - 4*x^2 - x + 1]

An example of a relative extension where the base field is not the maximal totally real subfield.

sage: E_0.<a> = NumberField(x^2 - 4*x + 16)

sage: y = polygen(E_0)

sage: E.<z> = E_0.extension(y^2 - E_0.gen() / 2)

sage: E.maximal_totally_real_subfield()

[Number Field in z1 with defining polynomial x^2 - 2*x - 5, Composite map:

From: Number Field in z1 with defining polynomial x^2 - 2*x - 5

To: Number Field in z with defining polynomial x^2 - 1/2*a over its base field

Defn: Ring morphism:

From: Number Field in z1 with defining polynomial x^2 - 2*x - 5

To: Number Field in z with defining polynomial x^4 - 2*x^3 + x^2 + 6*x + 3

Defn: z1 |--> -1/3*z^3 + 1/3*z^2 + z - 1

then

Isomorphism map:

From: Number Field in z with defining polynomial x^4 - 2*x^3 + x^2 + 6*x + 3

To: Number Field in z with defining polynomial x^2 - 1/2*a over its base field]

"""

try:

return self.__max_tot_real_sub

except(AttributeError):

pass

if isinstance(

self, sage.rings.number_field.number_field.NumberField_quadratic):

if self.discriminant() > 0:

self.__max_tot_real_sub = [self, self.coerce_map_from(self)]

return self.__max_tot_real_sub

else:

self.__max_tot_real_sub = [QQ, self.coerce_map_from(QQ)]

return self.__max_tot_real_sub

if isinstance(

self, sage.rings.number_field.number_field.NumberField_cyclotomic):

zeta = self.gen()

self.__max_tot_real_sub = self.subfield(zeta + zeta ** (-1))

return self.__max_tot_real_sub

if self.is_totally_real():

self.__max_tot_real_sub = [self, self.coerce_map_from(self)]

return self.__max_tot_real_sub

if self.is_absolute():

K = self

else:

if self.is_CM_extension():

self.__max_tot_real_sub = [self.base_field(), self.coerce_map_from(self.base_field())]

return self.__max_tot_real_sub

K = self.absolute_field('z')

d = K.absolute_degree()

divs = d.divisors()[1:-1]

divs.reverse()

for i in divs:

possibilities = K.subfields(i)

for F, phi, _ in possibilities:

if F.is_totally_real():

if self.is_relative():

phi = phi.post_compose(K.structure()[0])

self.__max_tot_real_sub = [F, phi]

return self.__max_tot_real_sub

self.__max_tot_real_sub = [QQ, self.coerce_map_from(QQ)]

return self.__max_tot_real_sub

def complex_embeddings(self, prec=53):

r"""

Return all homomorphisms of this number field into the approximate

complex field with precision prec.

This always embeds into an MPFR based complex field. If you

want embeddings into the 53-bit double precision, which is

faster, use ``self.embeddings(CDF)``.

EXAMPLES::

sage: k.<a> = NumberField(x^5 + x + 17)

sage: v = k.complex_embeddings()

sage: ls = [phi(k.0^2) for phi in v] ; ls # random order

[2.97572074038...,

-2.40889943716 + 1.90254105304*I,

-2.40889943716 - 1.90254105304*I,

0.921039066973 + 3.07553311885*I,

0.921039066973 - 3.07553311885*I]

sage: K.<a> = NumberField(x^3 + 2)

sage: ls = K.complex_embeddings() ; ls # random order

[

Ring morphism:

From: Number Field in a with defining polynomial x^3 + 2

To: Complex Double Field

Defn: a |--> -1.25992104989...,

Ring morphism:

From: Number Field in a with defining polynomial x^3 + 2

To: Complex Double Field

Defn: a |--> 0.629960524947 - 1.09112363597*I,

Ring morphism:

From: Number Field in a with defining polynomial x^3 + 2

To: Complex Double Field

Defn: a |--> 0.629960524947 + 1.09112363597*I

]

"""

CC = sage.rings.complex_field.ComplexField(prec)

return self.embeddings(CC)

def real_embeddings(self, prec=53):

r"""

Return all homomorphisms of this number field into the approximate

real field with precision prec.

If prec is 53 (the default), then the real double field is

used; otherwise the arbitrary precision (but slow) real field

is used. If you want embeddings into the 53-bit double

precision, which is faster, use ``self.embeddings(RDF)``.

.. NOTE::

This function uses finite precision real numbers.

In functions that should output proven results, one

could use ``self.embeddings(AA)`` instead.

EXAMPLES::

sage: K.<a> = NumberField(x^3 + 2)

sage: K.real_embeddings()

[

Ring morphism:

From: Number Field in a with defining polynomial x^3 + 2

To: Real Field with 53 bits of precision

Defn: a |--> -1.25992104989487

]

sage: K.real_embeddings(16)

[

Ring morphism:

From: Number Field in a with defining polynomial x^3 + 2

To: Real Field with 16 bits of precision

Defn: a |--> -1.260

]

sage: K.real_embeddings(100)

[

Ring morphism:

From: Number Field in a with defining polynomial x^3 + 2

To: Real Field with 100 bits of precision

Defn: a |--> -1.2599210498948731647672106073

]

As this is a numerical function, the number of embeddings

may be incorrect if the precision is too low::

sage: K = NumberField(x^2+2*10^1000*x + 10^2000+1, 'a')

sage: len(K.real_embeddings())

sage: len(K.real_embeddings(100))

sage: len(K.real_embeddings(10000))

sage: len(K.embeddings(AA))

"""

K = sage.rings.real_mpfr.RealField(prec)

return self.embeddings(K)

def specified_complex_embedding(self):

r"""

Returns the embedding of this field into the complex numbers which has

been specified.

Fields created with the ``QuadraticField`` or

``CyclotomicField`` constructors come with an implicit

embedding. To get one of these fields without the embedding, use

the generic ``NumberField`` constructor.

EXAMPLES::

sage: QuadraticField(-1, 'I').specified_complex_embedding()

Generic morphism:

From: Number Field in I with defining polynomial x^2 + 1

To: Complex Lazy Field

Defn: I -> 1*I

sage: QuadraticField(3, 'a').specified_complex_embedding()

Generic morphism:

From: Number Field in a with defining polynomial x^2 - 3

To: Real Lazy Field

Defn: a -> 1.732050807568878?

sage: CyclotomicField(13).specified_complex_embedding()

Generic morphism:

From: Cyclotomic Field of order 13 and degree 12

To: Complex Lazy Field

Defn: zeta13 -> 0.885456025653210? + 0.464723172043769?*I

Most fields don't implicitly have embeddings unless explicitly

specified::

sage: NumberField(x^2-2, 'a').specified_complex_embedding() is None

True

sage: NumberField(x^3-x+5, 'a').specified_complex_embedding() is None

True

sage: NumberField(x^3-x+5, 'a', embedding=2).specified_complex_embedding()

Generic morphism:

From: Number Field in a with defining polynomial x^3 - x + 5

To: Real Lazy Field

Defn: a -> -1.904160859134921?

sage: NumberField(x^3-x+5, 'a', embedding=CDF.0).specified_complex_embedding()

Generic morphism:

From: Number Field in a with defining polynomial x^3 - x + 5

To: Complex Lazy Field

Defn: a -> 0.952080429567461? + 1.311248044077123?*I

This function only returns complex embeddings::

sage: K.<a> = NumberField(x^2-2, embedding=Qp(7)(2).sqrt())

sage: K.specified_complex_embedding() is None

True

sage: K.gen_embedding()

3 + 7 + 2*7^2 + 6*7^3 + 7^4 + 2*7^5 + 7^6 + 2*7^7 + 4*7^8 + 6*7^9 + 6*7^10 + 2*7^11 + 7^12 + 7^13 + 2*7^15 + 7^16 + 7^17 + 4*7^18 + 6*7^19 + O(7^20)

sage: K.coerce_embedding()

Generic morphism:

From: Number Field in a with defining polynomial x^2 - 2

To: 7-adic Field with capped relative precision 20

Defn: a -> 3 + 7 + 2*7^2 + 6*7^3 + 7^4 + 2*7^5 + 7^6 + 2*7^7 + 4*7^8 + 6*7^9 + 6*7^10 + 2*7^11 + 7^12 + 7^13 + 2*7^15 + 7^16 + 7^17 + 4*7^18 + 6*7^19 + O(7^20)

"""

embedding = self.coerce_embedding()

if embedding is not None:

from sage.rings.real_mpfr import mpfr_prec_min

from sage.rings.complex_field import ComplexField

if ComplexField(mpfr_prec_min()).has_coerce_map_from(embedding.codomain()):

return embedding

def gen_embedding(self):

"""

If an embedding has been specified, return the image of the

generator under that embedding. Otherwise return None.

EXAMPLES::

sage: QuadraticField(-7, 'a').gen_embedding()

2.645751311064591?*I

sage: NumberField(x^2+7, 'a').gen_embedding() # None

"""

embedding = self.coerce_embedding()

if embedding is None:

return None

else:

return embedding(self.gen())

def algebraic_closure(self):

"""

Return the algebraic closure of self (which is QQbar).

EXAMPLES::

sage: K. = QuadraticField(-1)

sage: K.algebraic_closure()

Algebraic Field

sage: K.<a> = NumberField(x^3-2)

sage: K.algebraic_closure()

Algebraic Field

sage: K = CyclotomicField(23)

sage: K.algebraic_closure()

Algebraic Field

"""

return sage.rings.all.QQbar

@cached_method

def conductor(self, check_abelian=True):

r"""

Computes the conductor of the abelian field `K`.

If check_abelian is set to false and the field is not an

abelian extension of `\mathbb{Q}`, the output is not meaningful.

INPUT:

- ``check_abelian`` - a boolean (default: ``True``); check to see that this is an abelian extension of `\mathbb{Q}`

OUTPUT:

Integer which is the conductor of the field.

EXAMPLES::

sage: K = CyclotomicField(27)

sage: k = K.subfields(9)[0][0]

sage: k.conductor()

sage: K.<t> = NumberField(x^3+x^2-2*x-1)

sage: K.conductor()

sage: K.<t> = NumberField(x^3+x^2-36*x-4)

sage: K.conductor()

109

sage: K = CyclotomicField(48)

sage: k = K.subfields(16)[0][0]

sage: k.conductor()

sage: NumberField(x,'a').conductor()

sage: NumberField(x^8 - 8*x^6 + 19*x^4 - 12*x^2 + 1,'a').conductor()

sage: NumberField(x^8 + 7*x^4 + 1,'a').conductor()

sage: NumberField(x^8 - 40*x^6 + 500*x^4 - 2000*x^2 + 50,'a').conductor()

160

ALGORITHM:

For odd primes, it is easy to compute from the ramification

index because the p-Sylow subgroup is cyclic. For p=2, there

are two choices for a given ramification index. They can be

distinguished by the parity of the exponent in the discriminant

of a 2-adic completion.

"""

m = 1

if check_abelian and not self.is_abelian():

raise ValueError("The conductor is only defined for abelian fields")

try:

De = self.__disc

except AttributeError:

De = self.polynomial().discriminant()

A = De.numerator().prime_factors()+De.denominator().prime_factors()

else:

A = De.prime_factors()

for p in A:

R = self.maximal_order(p)

e = R.fractional_ideal(p).prime_factors()[0].ramification_index()

if e!= 1:

if p==2:

m *= e*2

c = R.discriminant().valuation(2)

c /= self.polynomial().degree()/e

if is_odd(c):

m *= 2

else:

m *= p**(e.valuation(p)+1)

return m

def latex_variable_name(self, name=None):

"""

Return the latex representation of the variable name for this

number field.

EXAMPLES::

sage: NumberField(x^2 + 3, 'a').latex_variable_name()

'a'

sage: NumberField(x^3 + 3, 'theta3').latex_variable_name()

'\\theta_{3}'

sage: CyclotomicField(5).latex_variable_name()

'\\zeta_{5}'

"""

if name is None:

return self.__latex_variable_name

else:

self.__latex_variable_name = name

def _repr_(self):

"""

Return string representation of this number field.

EXAMPLES::

sage: k.<a> = NumberField(x^13 - (2/3)*x + 3)

sage: k._repr_()

'Number Field in a with defining polynomial x^13 - 2/3*x + 3'

"""

return "Number Field in %s with defining polynomial %s"%(

self.variable_name(), self.polynomial())

def _latex_(self):

r"""

Return latex representation of this number field. This is viewed as

a polynomial quotient ring over a field.

EXAMPLES::

sage: k.<a> = NumberField(x^13 - (2/3)*x + 3)

sage: k._latex_()

'\\Bold{Q}[a]/(a^{13} - \\frac{2}{3} a + 3)'

sage: latex(k)

\Bold{Q}[a]/(a^{13} - \frac{2}{3} a + 3)

Numbered variables are often correctly typeset::

sage: k.<theta25> = NumberField(x^25+x+1)

sage: print(k._latex_())

\Bold{Q}[\theta_{25}]/(\theta_{25}^{25} + \theta_{25} + 1)

"""

return "%s[%s]/(%s)"%(latex(QQ), self.latex_variable_name(),

self.polynomial()._latex_(self.latex_variable_name()))

def _ideal_class_(self, n=0):

"""

Return the Python class used in defining the zero ideal of the ring

of integers of this number field.

This function is required by the general ring/ideal machinery. The

value defined here is the default value for all number fields.

EXAMPLES::

sage: NumberField(x^2 + 2, 'c')._ideal_class_()

"""

return sage.rings.number_field.number_field_ideal.NumberFieldIdeal

def _fractional_ideal_class_(self):

"""

Return the Python class used in defining fractional ideals of the

ring of integers of this number field.

This function is required by the general ring/ideal machinery. The

value defined here is the default value for all number fields

*except* relative number fields; this function is overridden by

one of the same name on class NumberField_relative.

EXAMPLES::

sage: NumberField(x^2 + 2, 'c')._fractional_ideal_class_()

"""

return sage.rings.number_field.number_field_ideal.NumberFieldFractionalIdeal

def ideal(self, *gens, **kwds):

"""

K.ideal() returns a fractional ideal of the field, except for the

zero ideal which is not a fractional ideal.

EXAMPLES::

sage: K.=NumberField(x^2+1)

sage: K.ideal(2)

Fractional ideal (2)

sage: K.ideal(2+i)

Fractional ideal (i + 2)

sage: K.ideal(0)

Ideal (0) of Number Field in i with defining polynomial x^2 + 1

"""

try:

return self.fractional_ideal(*gens, **kwds)

except ValueError:

return sage.rings.ring.Ring.ideal(self, gens, **kwds)

def fractional_ideal(self, *gens, **kwds):

r"""

Return the ideal in `\mathcal{O}_K` generated by gens.

This overrides the ``sage.rings.ring.Field`` method to

use the ``sage.rings.ring.Ring`` one instead, since

we're not really concerned with ideals in a field but in its ring

of integers.

INPUT:

- ``gens`` - a list of generators, or a number field

ideal.

EXAMPLES::

sage: K.<a> = NumberField(x^3-2)

sage: K.fractional_ideal([1/a])

Fractional ideal (1/2*a^2)

One can also input a number field ideal itself,

or, more usefully, for a tower of number fields an ideal

in one of the fields lower down the tower.

sage: K.fractional_ideal(K.ideal(a))

Fractional ideal (a)

sage: L. = K.extension(x^2 - 3, x^2 + 1)

sage: M.<c> = L.extension(x^2 + 1)

sage: L.ideal(K.ideal(2, a))

Fractional ideal (a)

sage: M.ideal(K.ideal(2, a)) == M.ideal(a*(b - c)/2)

True

The zero ideal is not a fractional ideal!

sage: K.fractional_ideal(0)

Traceback (most recent call last):

...

ValueError: gens must have a nonzero element (zero ideal is not a fractional ideal)

"""

if len(gens) == 1 and isinstance(gens[0], (list, tuple)):

gens = gens[0]

if len(gens) == 1 and isinstance(gens[0], NumberFieldFractionalIdeal):

I = gens[0]

if I.number_field() is self:

return I

else:

gens = I.gens()

return self._fractional_ideal_class_()(self, gens, **kwds)

def ideals_of_bdd_norm(self, bound):

"""

All integral ideals of bounded norm.

INPUT:

- ``bound`` - a positive integer

OUTPUT: A dict of all integral ideals I such that Norm(I) <= bound,

keyed by norm.

EXAMPLES::

sage: K.<a> = NumberField(x^2 + 23)

sage: d = K.ideals_of_bdd_norm(10)

sage: for n in d:

....: print(n)

....: for I in d[n]:

....: print(I)

Fractional ideal (1)

Fractional ideal (2, 1/2*a - 1/2)

Fractional ideal (2, 1/2*a + 1/2)

Fractional ideal (3, 1/2*a - 1/2)

Fractional ideal (3, 1/2*a + 1/2)

Fractional ideal (4, 1/2*a + 3/2)

Fractional ideal (2)

Fractional ideal (4, 1/2*a + 5/2)

Fractional ideal (1/2*a - 1/2)

Fractional ideal (6, 1/2*a + 5/2)

Fractional ideal (6, 1/2*a + 7/2)

Fractional ideal (1/2*a + 1/2)

Fractional ideal (1/2*a + 3/2)

Fractional ideal (4, a - 1)

Fractional ideal (4, a + 1)

Fractional ideal (1/2*a - 3/2)

Fractional ideal (9, 1/2*a + 11/2)

Fractional ideal (3)

Fractional ideal (9, 1/2*a + 7/2)

"""

hnf_ideals = self.pari_nf().ideallist(bound)

d = {}

for i in range(bound):

d[i+1] = [self.ideal(hnf) for hnf in hnf_ideals[i]]

return d

def primes_above(self, x, degree=None):

r"""

Return prime ideals of self lying over x.

INPUT:

- ``x``: usually an element or ideal of self. It

should be such that self.ideal(x) is sensible. This excludes x=0.

- ``degree`` (default: None): None or an integer.

If None, find all primes above x of any degree. If an integer, find

all primes above x such that the resulting residue field has

exactly this degree.

OUTPUT: A list of prime ideals of self lying over x. If degree

is specified and no such ideal exists, returns the empty list.

The output is sorted by residue degree first, then by

underlying prime (or equivalently, by norm).

EXAMPLES::

sage: x = ZZ['x'].gen()

sage: F.<t> = NumberField(x^3 - 2)

sage: P2s = F.primes_above(2)

sage: P2s # random

[Fractional ideal (-t)]

sage: all(2 in P2 for P2 in P2s)

True

sage: all(P2.is_prime() for P2 in P2s)

True

sage: [ P2.norm() for P2 in P2s ]

[2]

sage: P3s = F.primes_above(3)

sage: P3s # random

[Fractional ideal (t + 1)]

sage: all(3 in P3 for P3 in P3s)

True

sage: all(P3.is_prime() for P3 in P3s)

True

sage: [ P3.norm() for P3 in P3s ]

[3]

The ideal (3) is totally ramified in F, so there is no degree 2

prime above 3::

sage: F.primes_above(3, degree=2)

[]

sage: [ id.residue_class_degree() for id, _ in F.ideal(3).factor() ]

[1]

Asking for a specific degree works::

sage: P5_1s = F.primes_above(5, degree=1)

sage: P5_1s # random

[Fractional ideal (-t^2 - 1)]

sage: P5_1 = P5_1s[0]; P5_1.residue_class_degree()

sage: P5_2s = F.primes_above(5, degree=2)

sage: P5_2s # random

[Fractional ideal (t^2 - 2*t - 1)]

sage: P5_2 = P5_2s[0]; P5_2.residue_class_degree()

Works in relative extensions too::

sage: PQ.<X> = QQ[]

sage: F.<a, b> = NumberField([X^2 - 2, X^2 - 3])

sage: PF.<Y> = F[]

sage: K.<c> = F.extension(Y^2 - (1 + a)*(a + b)*a*b)

sage: I = F.ideal(a + 2*b)

sage: P, Q = K.primes_above(I)

sage: K.ideal(I) == P^4*Q

True

sage: K.primes_above(I, degree=1) == [P]

True

sage: K.primes_above(I, degree=4) == [Q]

True

It doesn't make sense to factor the ideal (0), so this raises an error::

sage: F.prime_above(0)

Traceback (most recent call last):

...

AttributeError: 'NumberFieldIdeal' object has no attribute 'prime_factors'

"""

if degree is not None:

degree = ZZ(degree)

facs = sorted([ (id.residue_class_degree(), id.absolute_norm(), id) for id in self.prime_factors(x) ])

if degree is None:

return [ id for d, n, id in facs ]

else:

return [ id for d, n, id in facs if d == degree ]

def prime_above(self, x, degree=None):

r"""

Return a prime ideal of self lying over x.

INPUT:

- ``x``: usually an element or ideal of self. It

should be such that self.ideal(x) is sensible. This excludes x=0.

- ``degree`` (default: None): None or an integer.

If one, find a prime above x of any degree. If an integer, find a

prime above x such that the resulting residue field has exactly

this degree.

OUTPUT: A prime ideal of self lying over x. If degree is specified

and no such ideal exists, raises a ValueError.

EXAMPLES::

sage: x = ZZ['x'].gen()

sage: F.<t> = NumberField(x^3 - 2)

sage: P2 = F.prime_above(2)

sage: P2 # random

Fractional ideal (-t)

sage: 2 in P2

True

sage: P2.is_prime()

True

sage: P2.norm()

sage: P3 = F.prime_above(3)

sage: P3 # random

Fractional ideal (t + 1)

sage: 3 in P3

True

sage: P3.is_prime()

True

sage: P3.norm()

The ideal (3) is totally ramified in F, so there is no degree 2

prime above 3::

sage: F.prime_above(3, degree=2)

Traceback (most recent call last):

...

ValueError: No prime of degree 2 above Fractional ideal (3)

sage: [ id.residue_class_degree() for id, _ in F.ideal(3).factor() ]

[1]

Asking for a specific degree works::

sage: P5_1 = F.prime_above(5, degree=1)

sage: P5_1 # random

Fractional ideal (-t^2 - 1)

sage: P5_1.residue_class_degree()

sage: P5_2 = F.prime_above(5, degree=2)

sage: P5_2 # random

Fractional ideal (t^2 - 2*t - 1)

sage: P5_2.residue_class_degree()

Relative number fields are ok::

sage: G = F.extension(x^2 - 11, 'b')

sage: G.prime_above(7)

Fractional ideal (b + 2)

It doesn't make sense to factor the ideal (0)::

sage: F.prime_above(0)

Traceback (most recent call last):

...

AttributeError: 'NumberFieldIdeal' object has no attribute 'prime_factors'

"""

ids = self.primes_above(x, degree)

if not ids:

raise ValueError("No prime of degree %s above %s" % (degree, self.ideal(x)))

return ids[0]

def primes_of_bounded_norm(self, B):

r"""

Returns a sorted list of all prime ideals with norm at most `B`.

INPUT:

- ``B`` -- a positive integer or real; upper bound on the norms of the

primes generated.

OUTPUT:

A list of all prime ideals of this number field of norm at

most `B`, sorted by norm. Primes of the same norm are sorted

using the comparison function for ideals, which is based on

the Hermite Normal Form.

.. note::

See also :meth:`primes_of_bounded_norm_iter` for an

iterator version of this, but note that the iterator sorts

the primes in order of underlying rational prime, not by

norm.

EXAMPLES::

sage: K. = QuadraticField(-1)

sage: K.primes_of_bounded_norm(10)

[Fractional ideal (i + 1), Fractional ideal (-i - 2), Fractional ideal (2*i + 1), Fractional ideal (3)]

sage: K.primes_of_bounded_norm(1)

[]

sage: K.<a> = NumberField(x^3-2)

sage: P = K.primes_of_bounded_norm(30)

sage: P

[Fractional ideal (a),

Fractional ideal (a + 1),

Fractional ideal (-a^2 - 1),

Fractional ideal (a^2 + a - 1),

Fractional ideal (2*a + 1),

Fractional ideal (-2*a^2 - a - 1),

Fractional ideal (a^2 - 2*a - 1),

Fractional ideal (a + 3)]

sage: [p.norm() for p in P]

[2, 3, 5, 11, 17, 23, 25, 29]

"""

try:

B = ZZ(B)

except (TypeError, AttributeError):

try:

B = ZZ(B.ceil())

except (TypeError, AttributeError):

raise TypeError("%s is not valid bound on prime ideals" % B)

if B<2:

return []

if self is QQ:

return arith.primes(B+1)

else:

P = [pp for p in arith.primes(B+1) for pp in self.primes_above(p)]

P = [p for p in P if p.norm() <= B]

P.sort(key=lambda P: (P.norm(),P))

return P

def primes_of_bounded_norm_iter(self, B):

r"""

Iterator yielding all prime ideals with norm at most `B`.

INPUT:

- ``B`` -- a positive integer or real; upper bound on the norms of the

primes generated.

OUTPUT:

An iterator over all prime ideals of this number field of norm

at most `B`.

.. note::

The output is not sorted by norm, but by size of the

underlying rational prime.

EXAMPLES::

sage: K. = QuadraticField(-1)

sage: it = K.primes_of_bounded_norm_iter(10)

sage: list(it)

[Fractional ideal (i + 1),

Fractional ideal (3),

Fractional ideal (-i - 2),

Fractional ideal (2*i + 1)]

sage: list(K.primes_of_bounded_norm_iter(1))

[]

"""

try:

B = ZZ(B)

except (TypeError, AttributeError):

try:

B = ZZ(B.ceil())

except (TypeError, AttributeError):

raise TypeError("%s is not valid bound on prime ideals" % B)

if B < 2:

return

if self is QQ:

for p in arith.primes(B+1):

yield p

else:

for p in arith.primes(B+1):

for pp in self.primes_above(p):

if pp.norm() <= B:

yield pp

def primes_of_degree_one_iter(self, num_integer_primes=10000, max_iterations=100):

r"""

Return an iterator yielding prime ideals of absolute degree one and

small norm.

.. warning::

It is possible that there are no primes of `K` of

absolute degree one of small prime norm, and it possible

that this algorithm will not find any primes of small norm.

See module :mod:`sage.rings.number_field.small_primes_of_degree_one`

for details.

INPUT:

- ``num_integer_primes (default: 10000)`` - an

integer. We try to find primes of absolute norm no greater than the

num_integer_primes-th prime number. For example, if

num_integer_primes is 2, the largest norm found will be 3, since

the second prime is 3.

- ``max_iterations (default: 100)`` - an integer. We

test max_iterations integers to find small primes before raising

StopIteration.

EXAMPLES::

sage: K.<z> = CyclotomicField(10)

sage: it = K.primes_of_degree_one_iter()

sage: Ps = [ next(it) for i in range(3) ]

sage: Ps # random

[Fractional ideal (z^3 + z + 1), Fractional ideal (3*z^3 - z^2 + z - 1), Fractional ideal (2*z^3 - 3*z^2 + z - 2)]

sage: [ P.norm() for P in Ps ] # random

[11, 31, 41]

sage: [ P.residue_class_degree() for P in Ps ]

[1, 1, 1]

"""

from sage.rings.number_field.small_primes_of_degree_one import Small_primes_of_degree_one_iter

return Small_primes_of_degree_one_iter(self, num_integer_primes, max_iterations)

def primes_of_degree_one_list(self, n, num_integer_primes=10000, max_iterations=100):

r"""

Return a list of n prime ideals of absolute degree one and small

norm.

.. warning::

It is possible that there are no primes of `K` of

absolute degree one of small prime norm, and it possible

that this algorithm will not find any primes of small norm.

See module :mod:`sage.rings.number_field.small_primes_of_degree_one`

for details.

INPUT:

- ``num_integer_primes (default: 10000)`` - an

integer. We try to find primes of absolute norm no greater than the

num_integer_primes-th prime number. For example, if

num_integer_primes is 2, the largest norm found will be 3, since

the second prime is 3.

- ``max_iterations (default: 100)`` - an integer. We

test max_iterations integers to find small primes before raising

StopIteration.

EXAMPLES::

sage: K.<z> = CyclotomicField(10)

sage: Ps = K.primes_of_degree_one_list(3)

sage: Ps # random output

[Fractional ideal (-z^3 - z^2 + 1), Fractional ideal (2*z^3 - 2*z^2 + 2*z - 3), Fractional ideal (2*z^3 - 3*z^2 + z - 2)]

sage: [ P.norm() for P in Ps ]

[11, 31, 41]

sage: [ P.residue_class_degree() for P in Ps ]

[1, 1, 1]

"""

it = self.primes_of_degree_one_iter()

return [ next(it) for i in range(n) ]

def completely_split_primes(self, B = 200):

r"""

Returns a list of rational primes which split completely in the number field `K`.

INPUT:

- ``B`` -- a positive integer bound (default: 200)

OUTPUT:

A list of all primes ``p < B`` which split completely in ``K``.

EXAMPLES::

sage: K.<xi> = NumberField(x^3 - 3*x + 1)

sage: K.completely_split_primes(100)

[17, 19, 37, 53, 71, 73, 89]

"""

from sage.rings.fast_arith import prime_range

from sage.rings.finite_rings.finite_field_constructor import GF

from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing

from sage.arith.all import factor

split_primes = []

for p in prime_range(B):

Fp = GF(p)

FpT = PolynomialRing(Fp,'T')

g = FpT(self.defining_polynomial())

if len(factor(g)) == self.degree():

split_primes.append(p)

return split_primes

def _is_valid_homomorphism_(self, codomain, im_gens):

"""

Return whether or not there is a homomorphism defined by the given

images of generators.

To do this we just check that the elements of the image of the

given generator (im_gens always has length 1) satisfies the

relation of the defining poly of this field.

EXAMPLES::

sage: k.<a> = NumberField(x^2 - 3)

sage: k._is_valid_homomorphism_(QQ, [0])

False

sage: k._is_valid_homomorphism_(k, [])

False

sage: k._is_valid_homomorphism_(k, [a])

True

sage: k._is_valid_homomorphism_(k, [-a])

True

sage: k._is_valid_homomorphism_(k, [a+1])

False

"""

try:

if len(im_gens) != 1:

return False

# We need that elements of the base ring of the polynomial

# ring map canonically into codomain.

codomain._coerce_(QQ.one())

f = self.defining_polynomial()

return codomain(f(im_gens[0])) == 0

except (TypeError, ValueError):

return False

@cached_method

def _pari_absolute_structure(self):

r"""

Return data relating the Sage and PARI absolute polynomials.

OUTPUT:

Let `L` be this number field, and let `f` be the defining

polynomial of `K` over `\QQ`. This method returns a triple

``(g, alpha, beta)``, where

- ``g`` is the defining relative polynomial of the PARI ``nf``

structure (see :meth:`pari_nf`);

- ``alpha`` is the image of `x \bmod f` under some isomorphism

`\phi\colon K[x]/(f) \to K[x]/(g)`

- ``beta`` is the image of `x \bmod g` under the inverse

isomorphism `\phi^{-1}\colon K[x]/(g) \to K[x]/(f)`

EXAMPLES::

If `f` is monic and integral, the result satisfies ``g = f``

and ``alpha = beta = x``::

sage: K.<a> = NumberField(x^2 - 2)

sage: K._pari_absolute_structure()

(y^2 - 2, Mod(y, y^2 - 2), Mod(y, y^2 - 2))

An example where `f` neither monic nor integral::

sage: K.<a> = NumberField(2*x^2 + 1/3)

sage: K._pari_absolute_structure()

(y^2 + 6, Mod(1/6*y, y^2 + 6), Mod(6*y, y^2 + 1/6))

"""

f = self.absolute_polynomial()._pari_with_name('y')

if f.pollead() == f.content().denominator() == 1:

g = f

alpha = beta = g.variable().Mod(g)

else:

g, alpha = f.polredbest(flag=1)

beta = alpha.modreverse()

return g, alpha, beta

def pari_polynomial(self, name='x'):

"""

Return the PARI polynomial corresponding to this number field.

INPUT:

- ``name`` -- variable name (default: ``'x'``)

OUTPUT:

A monic polynomial with integral coefficients (PARI ``t_POL``)

defining the PARI number field corresponding to ``self``.

.. WARNING::

This is *not* the same as simply converting the defining

polynomial to PARI.

EXAMPLES::

sage: y = polygen(QQ)

sage: k.<a> = NumberField(y^2 - 3/2*y + 5/3)

sage: k.pari_polynomial()

x^2 - x + 40

sage: k.polynomial().__pari__()

x^2 - 3/2*x + 5/3

sage: k.pari_polynomial('a')

a^2 - a + 40

Some examples with relative number fields::

sage: k.<a, c> = NumberField([x^2 + 3, x^2 + 1])

sage: k.pari_polynomial()

x^4 + 8*x^2 + 4

sage: k.pari_polynomial('a')

a^4 + 8*a^2 + 4

sage: k.absolute_polynomial()

x^4 + 8*x^2 + 4

sage: k.relative_polynomial()

x^2 + 3

sage: k.<a, c> = NumberField([x^2 + 1/3, x^2 + 1/4])

sage: k.pari_polynomial()

x^4 - x^2 + 1

sage: k.absolute_polynomial()

x^4 - x^2 + 1

This fails with arguments which are not a valid PARI variable name::

sage: k = QuadraticField(-1)

sage: k.pari_polynomial('I')

Traceback (most recent call last):

...

PariError: I already exists with incompatible valence

sage: k.pari_polynomial('i')

i^2 + 1

sage: k.pari_polynomial('theta')

Traceback (most recent call last):

...

PariError: theta already exists with incompatible valence

"""

return self._pari_absolute_structure()[0].change_variable_name(name)

def pari_nf(self, important=True):

"""

Return the PARI number field corresponding to this field.

INPUT:

- ``important`` -- boolean (default: ``True``). If ``False``,

raise a ``RuntimeError`` if we need to do a difficult

discriminant factorization. This is useful when an integral

basis is not strictly required, such as for factoring

polynomials over this number field.

OUTPUT:

The PARI number field obtained by calling the PARI function

:pari:`nfinit` with ``self.pari_polynomial('y')`` as argument.

.. NOTE::

This method has the same effect as ``pari(self)``.

EXAMPLES::

sage: k.<a> = NumberField(x^4 - 3*x + 7); k

Number Field in a with defining polynomial x^4 - 3*x + 7

sage: k.pari_nf()[:4]

[y^4 - 3*y + 7, [0, 2], 85621, 1]

sage: pari(k)[:4]

[y^4 - 3*y + 7, [0, 2], 85621, 1]

sage: k.<a> = NumberField(x^4 - 3/2*x + 5/3); k

Number Field in a with defining polynomial x^4 - 3/2*x + 5/3

sage: k.pari_nf()

[y^4 - 324*y + 2160, [0, 2], 48918708, 216, ..., [36, 36*y, y^3 + 6*y^2 - 252, 6*y^2], [1, 0, 0, 252; 0, 1, 0, 0; 0, 0, 0, 36; 0, 0, 6, -36], [1, 0, 0, 0, 0, 0, -18, 42, 0, -18, -46, -60, 0, 42, -60, -60; 0, 1, 0, 0, 1, 0, 2, 0, 0, 2, -11, -1, 0, 0, -1, 9; 0, 0, 1, 0, 0, 0, 6, 6, 1, 6, -5, 0, 0, 6, 0, 0; 0, 0, 0, 1, 0, 6, -6, -6, 0, -6, -1, 2, 1, -6, 2, 0]]

sage: pari(k)

[y^4 - 324*y + 2160, [0, 2], 48918708, 216, ...]

sage: gp(k)

[y^4 - 324*y + 2160, [0, 2], 48918708, 216, ...]

With ``important=False``, we simply bail out if we cannot

easily factor the discriminant::

sage: p = next_prime(10^40); q = next_prime(10^41)

sage: K.<a> = NumberField(x^2 - p*q)

sage: K.pari_nf(important=False)

Traceback (most recent call last):

...

RuntimeError: Unable to factor discriminant with trial division

Next, we illustrate the ``maximize_at_primes`` and ``assume_disc_small``

parameters of the ``NumberField`` constructor. The following would take

a very long time without the ``maximize_at_primes`` option::

sage: K.<a> = NumberField(x^2 - p*q, maximize_at_primes=[p])

sage: K.pari_nf()

[y^2 - 100000000000000000000...]

Since the discriminant is square-free, this also works::

sage: K.<a> = NumberField(x^2 - p*q, assume_disc_small=True)

sage: K.pari_nf()

[y^2 - 100000000000000000000...]

"""

try:

return self._pari_nf

except AttributeError:

f = self.pari_polynomial("y")

if f.poldegree() > 1:

f = pari([f, self._pari_integral_basis(important=important)])

self._pari_nf = f.nfinit()

return self._pari_nf

def pari_zk(self):

"""

Integral basis of the PARI number field corresponding to this field.

This is the same as pari_nf().getattr('zk'), but much faster.

EXAMPLES::

sage: k.<a> = NumberField(x^3 - 17)

sage: k.pari_zk()

[1, 1/3*y^2 - 1/3*y + 1/3, y]

sage: k.pari_nf().getattr('zk')

[1, 1/3*y^2 - 1/3*y + 1/3, y]

"""

return self.pari_nf().nf_get_zk()

def __pari__(self):

"""

Return the PARI number field corresponding to this field.

EXAMPLES::

sage: k = NumberField(x^2 + x + 1, 'a')

sage: k.__pari__()

[y^2 + y + 1, [0, 1], -3, 1, ... [1, y], [1, 0; 0, 1], [1, 0, 0, -1; 0, 1, 1, -1]]

sage: pari(k)

[y^2 + y + 1, [0, 1], -3, 1, ...[1, y], [1, 0; 0, 1], [1, 0, 0, -1; 0, 1, 1, -1]]

"""

return self.pari_nf()

def _pari_init_(self):

"""

Return the PARI number field corresponding to this field.

EXAMPLES::

sage: k = NumberField(x^2 + x + 1, 'a')

sage: k._pari_init_()

'[y^2 + y + 1, [0, 1], -3, 1, ... [1, y], [1, 0; 0, 1], [1, 0, 0, -1; 0, 1, 1, -1]]'

sage: gp(k)

[y^2 + y + 1, [0, 1], -3, 1, ...[1, y], [1, 0; 0, 1], [1, 0, 0, -1; 0, 1, 1, -1]]

"""

return str(self.pari_nf())

def pari_bnf(self, proof=None, units=True):

"""

PARI big number field corresponding to this field.

INPUT:

- ``proof`` -- If False, assume GRH. If True, run PARI's

:pari:`bnfcertify` to make sure that the results are correct.

- ``units`` -- (default: True) If True, insist on having

fundamental units. If False, the units may or may not be

computed.

OUTPUT:

The PARI ``bnf`` structure of this number field.

.. warning::

Even with ``proof=True``, I wouldn't trust this to mean

that everything computed involving this number field is

actually correct.

EXAMPLES::

sage: k.<a> = NumberField(x^2 + 1); k

Number Field in a with defining polynomial x^2 + 1

sage: len(k.pari_bnf())

sage: k.pari_bnf()[:4]

[[;], matrix(0,3), [;], ...]

sage: len(k.pari_nf())

sage: k.<a> = NumberField(x^7 + 7); k

Number Field in a with defining polynomial x^7 + 7

sage: dummy = k.pari_bnf(proof=True)

"""

proof = get_flag(proof, "number_field")

# First compute bnf

try:

bnf = self._pari_bnf

except AttributeError:

f = self.pari_polynomial("y")

if units:

self._pari_bnf = f.bnfinit(1)

else:

self._pari_bnf = f.bnfinit()

bnf = self._pari_bnf

# Certify if needed

if proof and not getattr(self, "_pari_bnf_certified", False):

if bnf.bnfcertify() != 1:

raise ValueError("The result is not correct according to bnfcertify")

self._pari_bnf_certified = True

return bnf

def pari_rnfnorm_data(self, L, proof=True):

"""

Return the PARI :pari:`rnfisnorminit` data corresponding to the

extension L/self.

EXAMPLES::

sage: x = polygen(QQ)

sage: K = NumberField(x^2 - 2, 'alpha')

sage: L = K.extension(x^2 + 5, 'gamma')

sage: ls = K.pari_rnfnorm_data(L) ; len(ls)

sage: K.<a> = NumberField(x^2 + x + 1)

sage: P.<X> = K[]

sage: L. = NumberField(X^3 + a)

sage: ls = K.pari_rnfnorm_data(L); len(ls)

"""

if L.base_field() != self:

raise ValueError("L must be an extension of self")

Kbnf = self.pari_bnf(proof=proof)

return Kbnf.rnfisnorminit(L.pari_relative_polynomial())

def _gap_init_(self):

"""

Create a gap object representing self and return its name

EXAMPLES::

sage: z = QQ['z'].0

sage: K.<zeta> = NumberField(z^2 - 2)

sage: K._gap_init_() # the following variable name $sage1 represents the F.base_ring() in gap and is somehow random

'CallFuncList(function() local z,E; z:=Indeterminate($sage1,"z"); E:=AlgebraicExtension($sage1,z^2 - 2,"zeta"); return E; end,[])'

sage: k = gap(K)

sage: k

sage: k.GeneratorsOfDivisionRing()

[ zeta ]

The following tests that it is possible to use a defining

polynomial in the variable ``E``, even though by default

``E`` is used as a local variable in the above GAP

``CallFuncList``::

sage: P.<E> = QQ[]

sage: L.<tau> = NumberField(E^3 - 2)

sage: l = gap(L); l

sage: l.GeneratorsOfField()

[ tau ]

sage: gap(tau)^3

"""

if not self.is_absolute():

raise NotImplementedError("Currently, only simple algebraic extensions are implemented in gap")

G = sage.interfaces.gap.gap

q = self.polynomial()

if q.variable_name()!='E':

return 'CallFuncList(function() local %s,E; %s:=Indeterminate(%s,"%s"); E:=AlgebraicExtension(%s,%s,"%s"); return E; end,[])'%(q.variable_name(),q.variable_name(),G(self.base_ring()).name(),q.variable_name(),G(self.base_ring()).name(),repr(self.polynomial()),str(self.gen()))

else:

return 'CallFuncList(function() local %s,F; %s:=Indeterminate(%s,"%s"); F:=AlgebraicExtension(%s,%s,"%s"); return F; end,[])'%(q.variable_name(),q.variable_name(),G(self.base_ring()).name(),q.variable_name(),G(self.base_ring()).name(),repr(self.polynomial()),str(self.gen()))

def characteristic(self):

"""

Return the characteristic of this number field, which is of course

EXAMPLES::

sage: k.<a> = NumberField(x^99 + 2); k

Number Field in a with defining polynomial x^99 + 2

sage: k.characteristic()

"""

return ZZ.zero()

def class_group(self, proof=None, names='c'):

r"""

Return the class group of the ring of integers of this number

field.

INPUT:

- ``proof`` - if True then compute the class group

provably correctly. Default is True. Call number_field_proof to

change this default globally.

- ``names`` - names of the generators of this class

group.

OUTPUT: The class group of this number field.

EXAMPLES::

sage: K.<a> = NumberField(x^2 + 23)

sage: G = K.class_group(); G

Class group of order 3 with structure C3 of Number Field in a with defining polynomial x^2 + 23

sage: G.0

Fractional ideal class (2, 1/2*a - 1/2)

sage: G.gens()

(Fractional ideal class (2, 1/2*a - 1/2),)

sage: G.number_field()

Number Field in a with defining polynomial x^2 + 23

sage: G is K.class_group()

True

sage: G is K.class_group(proof=False)

False

sage: G.gens()

(Fractional ideal class (2, 1/2*a - 1/2),)

There can be multiple generators::

sage: k.<a> = NumberField(x^2 + 20072)

sage: G = k.class_group(); G

Class group of order 76 with structure C38 x C2 of Number Field in a with defining polynomial x^2 + 20072

sage: G.0 # random

Fractional ideal class (41, a + 10)

sage: G.0^38

Trivial principal fractional ideal class

sage: G.1 # random

Fractional ideal class (2, -1/2*a)

sage: G.1^2

Trivial principal fractional ideal class

Class groups of Hecke polynomials tend to be very small::

sage: f = ModularForms(97, 2).T(2).charpoly()

sage: f.factor()

(x - 3) * (x^3 + 4*x^2 + 3*x - 1) * (x^4 - 3*x^3 - x^2 + 6*x - 1)

sage: [NumberField(g,'a').class_group().order() for g,_ in f.factor()]

[1, 1, 1]

"""

proof = proof_flag(proof)

try:

return self.__class_group[proof, names]

except KeyError:

pass

except AttributeError:

self.__class_group = {}

k = self.pari_bnf(proof)

cycle_structure = tuple( ZZ(c) for c in k.bnf_get_cyc() )

# Gens is a list of ideals (the generators)

gens = tuple( self.ideal(hnf) for hnf in k.bnf_get_gen() )

G = ClassGroup(cycle_structure, names, self, gens, proof=proof)

self.__class_group[proof, names] = G

return G

def class_number(self, proof=None):

"""

Return the class number of this number field, as an integer.

INPUT:

- ``proof`` - bool (default: True unless you called

number_field_proof)

EXAMPLES::

sage: NumberField(x^2 + 23, 'a').class_number()

sage: NumberField(x^2 + 163, 'a').class_number()

sage: NumberField(x^3 + x^2 + 997*x + 1, 'a').class_number(proof=False)

1539

"""

proof = proof_flag(proof)

return self.class_group(proof).order()

def S_class_group(self, S, proof=None, names='c'):

"""

Returns the S-class group of this number field over its base field.

INPUT:

- ``S`` - a set of primes of the base field

- ``proof`` - if False, assume the GRH in computing the class group.

Default is True. Call ``number_field_proof`` to change this

default globally.

- ``names`` - names of the generators of this class group.

OUTPUT:

The S-class group of this number field.

EXAMPLES:

A well known example::

sage: K.<a> = QuadraticField(-5)

sage: K.S_class_group([])

S-class group of order 2 with structure C2 of Number Field in a with defining polynomial x^2 + 5

When we include the prime `(2, a+1)`, the S-class group becomes

trivial::

sage: K.S_class_group([K.ideal(2,a+1)])

S-class group of order 1 of Number Field in a with defining polynomial x^2 + 5

TESTS::

sage: K.<a> = QuadraticField(-14)

sage: I = K.ideal(2,a)

sage: S = (I,)

sage: CS = K.S_class_group(S);CS

S-class group of order 2 with structure C2 of Number Field in a with defining polynomial x^2 + 14

sage: T = tuple([])

sage: CT = K.S_class_group(T);CT

S-class group of order 4 with structure C4 of Number Field in a with defining polynomial x^2 + 14

sage: K.class_group()

Class group of order 4 with structure C4 of Number Field in a with defining polynomial x^2 + 14

"""

proof = proof_flag(proof)

if all(P.is_principal() for P in S):

C = self.class_group(proof=proof)

Slist = list(zip([g.ideal() for g in C.gens()], C.invariants()))

else:

Slist = self._S_class_group_and_units(tuple(S), proof=proof)[1]

return SClassGroup(tuple(s[1] for s in Slist), names, self,

tuple(s[0] for s in Slist), tuple(S))

def S_units(self, S, proof=True):

"""

Returns a list of generators of the S-units.

INPUT:

- ``S`` -- a set of primes of the base field

- ``proof`` -- if ``False``, assume the GRH in computing the class group

OUTPUT:

A list of generators of the unit group.

.. note::

For more functionality see the S_unit_group() function.

EXAMPLES::

sage: K.<a> = QuadraticField(-3)

sage: K.unit_group()

Unit group with structure C6 of Number Field in a with defining polynomial x^2 + 3

sage: K.S_units([]) # random

[1/2*a + 1/2]

sage: K.S_units([])[0].multiplicative_order()

An example in a relative extension (see :trac:`8722`)::

sage: L.<a,b> = NumberField([x^2 + 1, x^2 - 5])

sage: p = L.ideal((-1/2*b - 1/2)*a + 1/2*b - 1/2)

sage: W = L.S_units([p]); [x.norm() for x in W]

[9, 1, 1]

Our generators should have the correct parent (:trac:`9367`)::

sage: _.<x> = QQ[]

sage: L.<alpha> = NumberField(x^3 + x + 1)

sage: p = L.S_units([ L.ideal(7) ])

sage: p[0].parent()

Number Field in alpha with defining polynomial x^3 + x + 1

TESTS:

This checks that the multiple entries issue at :trac:`9341` is fixed::

sage: _.<t> = QQ[]

sage: K.<T> = NumberField(t-1)

sage: I = K.ideal(2)

sage: K.S_units([I])

[2, -1]

sage: J = K.ideal(-2)

sage: K.S_units([I, J, I])

[2, -1]

"""

return self._S_class_group_and_units(tuple(S), proof=proof)[0]

@cached_method

def _S_class_group_and_units(self, S, proof=True):

"""

Compute S class group and units.

INPUT:

- ``S`` - a tuple of prime ideals of self

- ``proof`` - if False, assume the GRH in computing the class group

OUTPUT:

- ``units, clgp_gens``, where:

- ``units`` - A list of generators of the unit group.

- ``clgp_gens`` - A list of generators of the `S`-class group.

Each generator is represented as a pair ``(gen, order)``,

where ``gen`` is a fractional ideal of self and ``order`` is

its order in the `S`-class group.

EXAMPLES::

sage: K.<a> = NumberField(x^2+5)

sage: K._S_class_group_and_units(())

([-1], [(Fractional ideal (2, a + 1), 2)])

sage: K.<a> = NumberField(polygen(QQ))

sage: K._S_class_group_and_units( (K.ideal(5),) )

([5, -1], [])

TESTS::

sage: K.<a> = NumberField(x^3 - 381 * x + 127)

sage: K._S_class_group_and_units(tuple(K.primes_above(13)))

([2/13*a^2 + 1/13*a - 677/13,

1/13*a^2 + 7/13*a - 332/13,

-1/13*a^2 + 6/13*a + 345/13,

-1,

2/13*a^2 + 1/13*a - 755/13,

1/13*a^2 - 19/13*a - 7/13],

[(Fractional ideal (11, a - 2), 2), (Fractional ideal (19, a + 7), 2)])

Number fields defined by non-monic and non-integral

polynomials are supported (:trac:`252`)::

sage: K.<a> = NumberField(2*x^2 - 1/3)

sage: K._S_class_group_and_units(tuple(K.primes_above(2) + K.primes_above(3)))

([-6*a + 2, 6*a + 3, -1, 12*a + 5], [])

"""

K_pari = self.pari_bnf(proof=proof)

from sage.misc.all import uniq

S_pari = [p.pari_prime() for p in uniq(S)]

result = K_pari.bnfsunit(S_pari)

units = [self(x, check=False) for x in result[0]] + self.unit_group().gens_values()

orders = result[4][1].sage()

gens = [self.ideal(_) for _ in result[4][2]]

return units, [(gens[k], orders[k]) for k in range(len(orders)) if orders[k] > 1]

@cached_method

def _S_class_group_quotient_matrix(self, S):

r"""

Return the matrix of the quotient map from the class group to the

S-class group. The result is cached.

EXAMPLES::

sage: K.<a> = QuadraticField(-21)

sage: K._S_class_group_quotient_matrix((K.ideal([2, a+1]),))

[1]

[0]

sage: K._S_class_group_quotient_matrix((K.ideal([5, a+2]),))

[0]

[1]

sage: K._S_class_group_quotient_matrix(())

[1 0]

[0 1]

sage: K.<a> = QuadraticField(-105)

sage: K._S_class_group_quotient_matrix((K.ideal(11, a + 4),))

[0 0]

[1 0]

[0 1]

"""

from sage.matrix.constructor import matrix

S_clgp_gens = self._S_class_group_and_units(S)[1]

a = len(S_clgp_gens)

c = self.class_group().ngens()

M = [u[0].ideal_class_log() for u in S_clgp_gens]

M += [x.ideal_class_log() for x in S]

M = matrix(ZZ, M)

A, Q = M.hermite_form(transformation=True)

assert A[:c] == 1 and A[c:] == 0

return Q[:c, :a]

def selmer_group(self, S, m, proof=True, orders=False):

r"""

Compute the group `K(S,m)`.

INPUT:

- ``S`` -- a set of primes of ``self``

- ``m`` -- a positive integer

- ``proof`` -- if False, assume the GRH in computing the class group

- ``orders`` (default False) -- if True, output two lists, the

generators and their orders

OUTPUT:

A list of generators of `K(S,m)`, and (optionally) their

orders as elements of `K^\times/(K^\times)^m`. This is the

subgroup of `K^\times/(K^\times)^m` consisting of elements `a`

such that the valuation of `a` is divisible by `m` at all

primes not in `S`. It fits in an exact sequence between the

units modulo `m`-th powers and the `m`-torsion in the

`S`-class group:

.. MATH::

1 \longrightarrow

O_{K,S}^\times / (O_{K,S}^\times)^m \longrightarrow

K(S,m) \longrightarrow

\operatorname{Cl}_{K,S}[m] \longrightarrow

The group `K(S,m)` contains the subgroup of those `a` such

that `K(\sqrt[m]{a})/K` is unramified at all primes of `K`

outside of `S`, but may contain it properly when not all

primes dividing `m` are in `S`.

EXAMPLES::

sage: K.<a> = QuadraticField(-5)

sage: K.selmer_group((), 2)

[-1, 2]

The previous example shows that the group generated by the

output may be strictly larger than the 'true' Selmer group of

elements giving extensions unramified outside `S`, since that

has order just 2, generated by `-1`::

sage: K.class_number()

sage: K.hilbert_class_field('b')

Number Field in b with defining polynomial x^2 + 1 over its base field

When `m` is prime all the orders are equal to `m`, but in general they are only divisors of `m`::

sage: K.<a> = QuadraticField(-5)

sage: P2 = K.ideal(2, -a+1)

sage: P3 = K.ideal(3, a+1)

sage: K.selmer_group((), 2, orders=True)

([-1, 2], [2, 2])

sage: K.selmer_group((), 4, orders=True)

([-1, 4], [2, 2])

sage: K.selmer_group([P2], 2)

[2, -1]

sage: K.selmer_group((P2,P3), 4)

[2, -a - 1, -1]

sage: K.selmer_group((P2,P3), 4, orders=True)

([2, -a - 1, -1], [4, 4, 2])

sage: K.selmer_group([P2], 3)

[2]

sage: K.selmer_group([P2, P3], 3)

[2, -a - 1]

sage: K.selmer_group([P2, P3, K.ideal(a)], 3) # random signs

[2, a + 1, a]

Example over `\QQ` (as a number field)::

sage: K.<a> = NumberField(polygen(QQ))

sage: K.selmer_group([],5)

[]

sage: K.selmer_group([K.prime_above(p) for p in [2,3,5]],2)

[2, 3, 5, -1]

sage: K.selmer_group([K.prime_above(p) for p in [2,3,5]],6, orders=True)

([2, 3, 5, -1], [6, 6, 6, 2])

TESTS::

sage: K.<a> = QuadraticField(-5)

sage: P2 = K.ideal(2, -a+1)

sage: P3 = K.ideal(3, a+1)

sage: P5 = K.ideal(a)

sage: S = K.selmer_group([P2, P3, P5], 3)

sage: S in ([2, a + 1, a], [2, a + 1, -a], [2, -a - 1, a], [2, -a - 1, -a]) or S

True

Verify that :trac:`14489` is fixed::

sage: K.<a> = NumberField(x^3 - 381 * x + 127)

sage: K.selmer_group(K.primes_above(13), 2)

[2/13*a^2 + 1/13*a - 677/13,

1/13*a^2 + 7/13*a - 332/13,

-1/13*a^2 + 6/13*a + 345/13,

-1,

2/13*a^2 + 1/13*a - 755/13,

1/13*a^2 - 19/13*a - 7/13,

2/13*a^2 + 53/13*a - 92/13,

2/13*a^2 + 40/13*a - 27/13]

Verify that :trac:`16708` is fixed::

sage: K.<a> = QuadraticField(-5)

sage: p = K.primes_above(2)[0]

sage: S = K.selmer_group((), 4)

sage: all(4.divides(x.valuation(p)) for x in S)

True

"""

units, clgp_gens = self._S_class_group_and_units(tuple(S), proof=proof)

gens = []

ords = []

for unit in units:

order = unit.multiplicative_order()

if order == Infinity:

gens.append(unit)

ords.append(m)

else:

m1 = order.gcd(m)

if m1!= 1:

gens.append(unit)

ords.append(m1)

card_S = len(S)

if card_S != 0:

from sage.matrix.constructor import Matrix

H = self.class_group()

gen_ords = [g.order() for g in H.gens()]

pari_ords = pari(gen_ords).Col()

Sords = [H(s).order() for s in S]

MS = Matrix(ZZ, [H(s).exponents() for s in S]).transpose()

pari_MS = pari(MS)

for gen, order in clgp_gens:

d = order.gcd(m)

if d != 1:

# The ideal I = gen^(order/d) has order d in Cl_S[m].

# After multiplying by primes in S, the ideal

# I^m = gen^(order*m/d) becomes principal. We take

# a generator of this ideal to get the corresponding

# generator of the m-Selmer group.

J = gen ** (order * m // d)

if card_S != 0 and not J.is_principal():

B = H(J).exponents()

pari_B = (-pari(B)).Col()

exps = pari_MS.matsolvemod(pari_ords, pari_B).Vec().sage()

Spart = prod([S[i] ** (exps[i] % Sords[i]) for i in range(card_S)])

J *= Spart

gens.append(self(J.gens_reduced()[0]))

ords.append(d)

if orders:

return gens, ords

else:

return gens

def selmer_group_iterator(self, S, m, proof=True):

r"""

Return an iterator through elements of the finite group `K(S,m)`.

INPUT:

- ``S`` -- a set of primes of ``self``

- ``m`` -- a positive integer

- ``proof`` -- if False, assume the GRH in computing the class group

OUTPUT:

An iterator yielding the distinct elements of `K(S,m)`. See

the docstring for :meth:`NumberField_generic.selmer_group` for

more information.

EXAMPLES::

sage: K.<a> = QuadraticField(-5)

sage: list(K.selmer_group_iterator((), 2))

[1, 2, -1, -2]

sage: list(K.selmer_group_iterator((), 4))

[1, 4, -1, -4]

sage: list(K.selmer_group_iterator([K.ideal(2, -a+1)], 2))

[1, -1, 2, -2]

sage: list(K.selmer_group_iterator([K.ideal(2, -a+1), K.ideal(3, a+1)], 2))

[1, -1, -a - 1, a + 1, 2, -2, -2*a - 2, 2*a + 2]

Examples over `\QQ` (as a number field)::

sage: K.<a> = NumberField(polygen(QQ))

sage: list(K.selmer_group_iterator([], 5))

[1]

sage: list(K.selmer_group_iterator([], 4))

[1, -1]

sage: list(K.selmer_group_iterator([K.prime_above(p) for p in [11,13]],2))

[1, -1, 13, -13, 11, -11, 143, -143]

"""

KSgens, ords = self.selmer_group(S=S, m=m, proof=proof, orders=True)

one = self.one()

from sage.misc.all import cartesian_product_iterator

for ev in cartesian_product_iterator([range(o) for o in ords]):

yield prod([p ** e for p, e in zip(KSgens, ev)], one)

def composite_fields(self, other, names=None, both_maps=False, preserve_embedding=True):

"""

Return the possible composite number fields formed from

``self`` and ``other``.

INPUT:

- ``other`` -- number field

- ``names`` -- generator name for composite fields

- ``both_maps`` -- boolean (default: ``False``)

- ``preserve_embedding`` -- boolean (default: True)

OUTPUT:

A list of the composite fields, possibly with maps.

If ``both_maps`` is ``True``, the list consists of quadruples

``(F, self_into_F, other_into_F, k)`` such that

``self_into_F`` is an embedding of ``self`` in ``F``,

``other_into_F`` is an embedding of in ``F``, and ``k`` is one

of the following:

- an integer such that ``F.gen()`` equals

``other_into_F(other.gen()) + k*self_into_F(self.gen())``;

- ``Infinity``, in which case ``F.gen()`` equals

``self_into_F(self.gen())``;

- ``None`` (when ``other`` is a relative number field).

If both ``self`` and ``other`` have embeddings into an ambient

field, then each ``F`` will have an embedding with respect to

which both ``self_into_F`` and ``other_into_F`` will be

compatible with the ambient embeddings.

If ``preserve_embedding`` is ``True`` and if ``self`` and

``other`` both have embeddings into the same ambient field, or

into fields which are contained in a common field, only the

compositum respecting both embeddings is returned. In all

other cases, all possible composite number fields are

returned.

EXAMPLES::

sage: K.<a> = NumberField(x^4 - 2)

sage: K.composite_fields(K)

[Number Field in a with defining polynomial x^4 - 2,

Number Field in a0 with defining polynomial x^8 + 28*x^4 + 2500]

A particular compositum is selected, together with compatible maps

into the compositum, if the fields are endowed with a real or

complex embedding::

sage: K1 = NumberField(x^4 - 2, 'a', embedding=RR(2^(1/4)))

sage: K2 = NumberField(x^4 - 2, 'a', embedding=RR(-2^(1/4)))

sage: K1.composite_fields(K2)

[Number Field in a with defining polynomial x^4 - 2]

sage: [F, f, g, k], = K1.composite_fields(K2, both_maps=True); F

Number Field in a with defining polynomial x^4 - 2

sage: f(K1.0), g(K2.0)

(a, -a)

With ``preserve_embedding`` set to ``False``, the embeddings

are ignored::

sage: K1.composite_fields(K2, preserve_embedding=False)

[Number Field in a with defining polynomial x^4 - 2,

Number Field in a0 with defining polynomial x^8 + 28*x^4 + 2500]

Changing the embedding selects a different compositum::

sage: K3 = NumberField(x^4 - 2, 'a', embedding=CC(2^(1/4)*I))

sage: [F, f, g, k], = K1.composite_fields(K3, both_maps=True); F

Number Field in a0 with defining polynomial x^8 + 28*x^4 + 2500

sage: f(K1.0), g(K3.0)

(1/240*a0^5 - 41/120*a0, 1/120*a0^5 + 19/60*a0)

If no embeddings are specified, the maps into the compositum

are chosen arbitrarily::

sage: Q1.<a> = NumberField(x^4 + 10*x^2 + 1)

sage: Q2. = NumberField(x^4 + 16*x^2 + 4)

sage: Q1.composite_fields(Q2, 'c')

[Number Field in c with defining polynomial x^8 + 64*x^6 + 904*x^4 + 3840*x^2 + 3600]

sage: F, Q1_into_F, Q2_into_F, k = Q1.composite_fields(Q2, 'c', both_maps=True)[0]

sage: Q1_into_F

Ring morphism:

From: Number Field in a with defining polynomial x^4 + 10*x^2 + 1

To: Number Field in c with defining polynomial x^8 + 64*x^6 + 904*x^4 + 3840*x^2 + 3600

Defn: a |--> 19/14400*c^7 + 137/1800*c^5 + 2599/3600*c^3 + 8/15*c

This is just one of four embeddings of ``Q1`` into ``F``::

sage: Hom(Q1, F).order()

Note that even with ``preserve_embedding=True``, this method may fail

to recognize that the two number fields have compatible embeddings, and

hence return several composite number fields::

sage: x = polygen(ZZ)

sage: A.<a> = NumberField(x^3 - 7, embedding=CC(-0.95+1.65*I))

sage: B.<a> = NumberField(x^9 - 7, embedding=QQbar.polynomial_root(x^9 - 7, RIF(1.2, 1.3)))

sage: len(A.composite_fields(B, preserve_embedding=True))

TESTS:

Let's check that embeddings are being respected::

sage: x = polygen(ZZ)

sage: K0. = CyclotomicField(7, 'a').subfields(3)[0][0].change_names()

sage: K1.<a1> = K0.extension(x^2 - 2*b^2, 'a1').absolute_field()

sage: K2.<a2> = K0.extension(x^2 - 3*b^2, 'a2').absolute_field()

We need embeddings, so we redefine::

sage: L1.<a1> = NumberField(K1.polynomial(), 'a1', embedding=CC.0)

sage: L2.<a2> = NumberField(K2.polynomial(), 'a2', embedding=CC.0)

sage: [CDF(a1), CDF(a2)]

[-0.6293842454258951, -0.7708351267200304]

and we get the same embeddings via the compositum::

sage: F, L1_into_F, L2_into_F, k = L1.composite_fields(L2, both_maps=True)[0]

sage: [CDF(L1_into_F(L1.gen())), CDF(L2_into_F(L2.gen()))]

[-0.6293842454258952, -0.7708351267200303]

Let's check that if only one field has an embedding, the resulting

fields do not have embeddings::

sage: L1.composite_fields(K2)[0].coerce_embedding() is None

True

sage: L2.composite_fields(K1)[0].coerce_embedding() is None

True

We check that other can be a relative number field::

sage: L.<a, b> = NumberField([x^3 - 5, x^2 + 3])

sage: CyclotomicField(3, 'w').composite_fields(L, both_maps=True)

[(Number Field in a with defining polynomial x^3 - 5 over its base field, Ring morphism:

From: Cyclotomic Field of order 3 and degree 2

To: Number Field in a with defining polynomial x^3 - 5 over its base field

Defn: w |--> -1/2*b - 1/2, Relative number field endomorphism of Number Field in a with defining polynomial x^3 - 5 over its base field

Defn: a |--> a

b |--> b, None)]

Number fields defined by non-monic and non-integral

polynomials are supported (:trac:`252`)::

sage: K.<a> = NumberField(x^2 + 1/2)

sage: L. = NumberField(3*x^2 - 1)

sage: K.composite_fields(L)

[Number Field in ab with defining polynomial 36*x^4 + 12*x^2 + 25]

sage: C = K.composite_fields(L, both_maps=True); C

[(Number Field in ab with defining polynomial 36*x^4 + 12*x^2 + 25,

Ring morphism:

From: Number Field in a with defining polynomial x^2 + 1/2

To: Number Field in ab with defining polynomial 36*x^4 + 12*x^2 + 25

Defn: a |--> -3/5*ab^3 - 7/10*ab,

Ring morphism:

From: Number Field in b with defining polynomial 3*x^2 - 1

To: Number Field in ab with defining polynomial 36*x^4 + 12*x^2 + 25

Defn: b |--> -3/5*ab^3 + 3/10*ab,

-1)]

sage: M, f, g, k = C[0]

sage: M.gen() == g(b) + k*f(a)

True

This also fixes the bugs reported at :trac:`14164` and

:trac:`18243`::

sage: R.<x> = QQ[]

sage: f = 6*x^5 + x^4 + x^2 + 5*x + 7

sage: r = f.roots(QQbar, multiplicities=False)

sage: F1 = NumberField(f.monic(), 'a', embedding=r[0])

sage: F2 = NumberField(f.monic(), 'a', embedding=r[1])

sage: (F, map1, map2, k) = F1.composite_fields(F2, both_maps=True)[0]

sage: F.degree()

sage: F.gen() == map2(F2.gen()) + k*map1(F1.gen())

True

sage: f = x^8 - 3*x^7 + 61/3*x^6 - 9*x^5 + 298*x^4 + 458*x^3 + 1875*x^2 + 4293*x + 3099

sage: F1 = NumberField(f, 'z', embedding=-1.18126721294295 + 3.02858651117832j)

sage: F2 = NumberField(f, 'z', embedding=-1.18126721294295 - 3.02858651117832j)

sage: (F, map1, map2, k) = F1.composite_fields(F2, both_maps=True)[0]

sage: F.degree()

sage: F.gen() == map2(F2.gen()) + k*map1(F1.gen())

True

Check that the bugs reported at :trac:`24357` are fixed::

sage: A.<a> = NumberField(x^9 - 7)

sage: B. = NumberField(x^3-7, embedding=a^3)

sage: C.<c> = QuadraticField(-1)

sage: B.composite_fields(C)

[Number Field in bc with defining polynomial x^6 + 3*x^4 + 14*x^3 + 3*x^2 - 42*x + 50]

sage: y = polygen(QQ, 'y')

sage: A.<a> = NumberField(x^3 - 7, embedding=CC(-0.95+1.65*I))

sage: B. = NumberField(y^9 - 7, embedding=CC(-1.16+0.42*I))

sage: A.composite_fields(B)

[Number Field in b with defining polynomial y^9 - 7]

"""

if not isinstance(other, NumberField_generic):

raise TypeError("other must be a number field.")

sv = self.variable_name(); ov = other.variable_name()

if names is None:

names = sv + (ov if ov != sv else "")

name = normalize_names(1, names)[0]

# should we try to preserve embeddings?

subfields_have_embeddings = preserve_embedding

if self.coerce_embedding() is None:

subfields_have_embeddings = False

if other.coerce_embedding() is None:

subfields_have_embeddings = False

if subfields_have_embeddings:

try:

from sage.categories.pushout import pushout

ambient_field = pushout(self.coerce_embedding().codomain(), other.coerce_embedding().codomain())

except sage.structure.coerce_exceptions.CoercionException:

ambient_field = None

if ambient_field is None:

subfields_have_embeddings = False

f = self.absolute_polynomial()

g = other.absolute_polynomial().change_variable_name(f.variable_name())

R = f.parent()

f = f.__pari__(); f /= f.content()

g = g.__pari__(); g /= g.content()

m = self.degree()

n = other.absolute_degree()

if not both_maps and not subfields_have_embeddings:

# short cut!

# eliminate duplicates from the fields given by polcompositum

# and return the resulting number fields. There is no need to

# check that the polynomials are irreducible.

C = []

for r in f.polcompositum(g):

if not any(r.nfisisom(s) for s in C):

C.append(r)

C = [R(_) for _ in C]

q = sum(1 for r in C if r.degree() != max(m, n))

if q == 1 and name != sv and name != ov:

names = [name]

else:

names = [name + str(i) for i in range(q)]

i = 0

rets = []

for r in C:

d = r.degree()

if d == m:

rets.append(self)

elif d == n:

rets.append(other)

else:

rets.append(NumberField(r, names[i], check=False))

i += 1

return rets

# If flag = 1, polcompositum outputs a vector of 4-component vectors

# [R, a, b, k], where R ranges through the list of all possible compositums

# as above, and a (resp. b) expresses the root of P (resp. Q) as

# an element of Q(X)/(R). Finally, k is a small integer such that

# b + ka = X modulo R.

# In this case duplicates must only be eliminated if embeddings are going

# to be preserved.

C = []

for v in f.polcompositum(g, 1):

if subfields_have_embeddings or not any(v[0].nfisisom(u[0]) for u in C):

C.append(v)

a = self.gen()

b = other.gen()

# If both subfields are provided with embeddings, then we must select

# the compositum which corresponds to these embeddings. We do this by

# evaluating the given polynomials at the corresponding embedded values.

# For the case we want, the result will be zero, but rounding errors are

# difficult to predict, so we just take the field which yields the

# minimum value.

if subfields_have_embeddings:

poly_vals = []

for r, _, _, k in C:

r = R(r)

k = ZZ(k)

embedding = other.coerce_embedding()(b) + k*self.coerce_embedding()(a)

poly_vals.append(r(embedding).abs())

i = poly_vals.index(min(poly_vals))

C = [C[i]]

q = sum(1 for r, _, _, _ in C if r.poldegree() != max(m, n))

if q == 1 and name != sv and name != ov:

names = [name, '']

else:

names = [name + str(i) for i in range(q + 1)]

if both_maps and not other.is_absolute():

other_abs = other.absolute_field('z')

from_other_abs, to_other_abs = other_abs.structure()

embedding = None

i = 0

rets = []

for r, a_in_F, b_in_F, k in C:

r = R(r)

d = r.degree()

if d == m and not both_maps:

rets.append(self)

elif d == n and not both_maps:

rets.append(other)

else:

k = ZZ(k)

if subfields_have_embeddings:

embedding = other.coerce_embedding()(b) + k*self.coerce_embedding()(a)

F = NumberField(r, names[i], check=False, embedding=embedding)

i += 1

if both_maps:

a_in_F = F(R(a_in_F.lift()))

b_in_F = F(R(b_in_F.lift()))

if other.is_absolute():

if d == m:

self_to_F = self.hom([self.gen()])

other_to_F = other.hom([(~self.hom([a_in_F]))(b_in_F)])

F = self

k = Infinity

i -= 1

elif d == n:

other_to_F = other.hom([other.gen()])

self_to_F = self.hom([(~other.hom([b_in_F]))(a_in_F)])

F = other

k = ZZ.zero()

i -= 1

else:

self_to_F = self.hom([a_in_F])

other_to_F = other.hom([b_in_F])

else:

other_abs_to_F = other_abs.hom([b_in_F])

other_to_F = RelativeNumberFieldHomomorphism_from_abs(other.Hom(F), other_abs_to_F*to_other_abs)

if d == m:

self_to_F = self.hom([self.gen()])

other_to_F = RelativeNumberFieldHomomorphism_from_abs(other.Hom(self), (~self.hom([a_in_F]))*other_abs_to_F*to_other_abs)

F = self

k = None

i -= 1

elif d == n:

other_to_F = RelativeNumberFieldHomomorphism_from_abs(other.Hom(other), from_other_abs)

self_to_F = self.hom([from_other_abs((~other_abs_to_F)(a_in_F))])

F = other

k = None

i -= 1

else:

self_to_F = self.hom([a_in_F])

other_to_F = RelativeNumberFieldHomomorphism_from_abs(other.Hom(F), other_abs_to_F*to_other_abs)

rets.append( (F, self_to_F, other_to_F, k) )

else:

rets.append(F)

return rets

def absolute_degree(self):

"""

Return the degree of self over `\QQ`.

EXAMPLES::

sage: NumberField(x^3 + x^2 + 997*x + 1, 'a').absolute_degree()

sage: NumberField(x + 1, 'a').absolute_degree()

sage: NumberField(x^997 + 17*x + 3, 'a', check=False).absolute_degree()

997

"""

return self.polynomial().degree()

def degree(self):

"""

Return the degree of this number field.

EXAMPLES::

sage: NumberField(x^3 + x^2 + 997*x + 1, 'a').degree()

sage: NumberField(x + 1, 'a').degree()

sage: NumberField(x^997 + 17*x + 3, 'a', check=False).degree()

997

"""

return self.polynomial().degree()

def different(self):

r"""

Compute the different fractional ideal of this number field.

The codifferent is the fractional ideal of all `x` in `K`

such that the trace of `xy` is an integer for

all `y \in O_K`.

The different is the integral ideal which is the inverse of

the codifferent.

See :wikipedia:`Different_ideal`

EXAMPLES::

sage: k.<a> = NumberField(x^2 + 23)

sage: d = k.different()

sage: d

Fractional ideal (-a)

sage: d.norm()

sage: k.disc()

-23

The different is cached::

sage: d is k.different()

True

Another example::

sage: k. = NumberField(x^2 - 123)

sage: d = k.different(); d

Fractional ideal (2*b)

sage: d.norm()

492

sage: k.disc()

492

"""

try:

return self.__different

except AttributeError:

self.__different = self.ideal(self.pari_nf().nf_get_diff())

return self.__different

def discriminant(self, v=None):

"""

Returns the discriminant of the ring of integers of the number

field, or if v is specified, the determinant of the trace pairing

on the elements of the list v.

INPUT:

- ``v`` -- (optional) list of elements of this number field

OUTPUT:

Integer if `v` is omitted, and Rational otherwise.

EXAMPLES::

sage: K.<t> = NumberField(x^3 + x^2 - 2*x + 8)

sage: K.disc()

-503

sage: K.disc([1, t, t^2])

-2012

sage: K.disc([1/7, (1/5)*t, (1/3)*t^2])

-2012/11025

sage: (5*7*3)^2

11025

sage: NumberField(x^2 - 1/2, 'a').discriminant()

"""

if v is None:

try:

return self.__disc

except AttributeError:

self.__disc = ZZ(self.pari_polynomial().nfdisc())

return self.__disc

else:

return QQ(self.trace_pairing(v).det())

def disc(self, v=None):

"""

Shortcut for self.discriminant.

EXAMPLES::

sage: k. = NumberField(x^2 - 123)

sage: k.disc()

492

"""

return self.discriminant(v=v)

def trace_dual_basis(self, b):

r"""

Compute the dual basis of a basis of ``self`` with respect to the trace pairing.

EXAMPLES::

sage: K.<a> = NumberField(x^3 + x + 1)

sage: b = [1, 2*a, 3*a^2]

sage: T = K.trace_dual_basis(b); T

[4/31*a^2 - 6/31*a + 13/31, -9/62*a^2 - 1/31*a - 3/31, 2/31*a^2 - 3/31*a + 4/93]

sage: [(b[i]*T[j]).trace() for i in range(3) for j in range(3)]

[1, 0, 0, 0, 1, 0, 0, 0, 1]

"""

if not len(b) == self.degree():

raise ValueError('Not a basis of the number field.')

M = self.trace_pairing(b)

if not M.is_invertible():

raise ValueError('Not a basis of the number field.')

return [sum([v[i]*b[i] for i in range(len(b))]) for v in M.inverse()]

def elements_of_norm(self, n, proof=None):

"""

Return a list of elements of norm ``n``.

INPUT:

- ``n`` -- integer in this number field

- ``proof`` -- boolean (default: ``True``, unless you called

``number_field_proof`` and set it otherwise)

OUTPUT:

A complete system of integral elements of norm `n`, modulo

units of positive norm.

EXAMPLES::

sage: K.<a> = NumberField(x^2+1)

sage: K.elements_of_norm(3)

[]

sage: K.elements_of_norm(50)

[-7*a + 1, 5*a - 5, 7*a + 1]

TESTS:

Number fields defined by non-monic and non-integral

polynomials are supported (:trac:`252`)::

sage: K.<a> = NumberField(7/9*x^3 + 7/3*x^2 - 56*x + 123)

sage: K.elements_of_norm(7)

[7/225*a^2 - 7/75*a - 42/25]

"""

proof = proof_flag(proof)

B = self.pari_bnf(proof).bnfisintnorm(n)

return [self(x, check=False) for x in B]

def extension(self, poly, name=None, names=None, *args, **kwds):

"""

Return the relative extension of this field by a given polynomial.

EXAMPLES::

sage: K.<a> = NumberField(x^3 - 2)

sage: R.<t> = K[]

sage: L. = K.extension(t^2 + a); L

Number Field in b with defining polynomial t^2 + a over its base field

We create another extension::

sage: k.<a> = NumberField(x^2 + 1); k

Number Field in a with defining polynomial x^2 + 1

sage: y = polygen(QQ,'y')

sage: m. = k.extension(y^2 + 2); m

Number Field in b with defining polynomial y^2 + 2 over its base field

Note that b is a root of `y^2 + 2`::

sage: b.minpoly()

x^2 + 2

sage: b.minpoly('z')

z^2 + 2

A relative extension of a relative extension::

sage: k.<a> = NumberField([x^2 + 1, x^3 + x + 1])

sage: R.<z> = k[]

sage: L. = NumberField(z^3 + 3 + a); L

Number Field in b with defining polynomial z^3 + a0 + 3 over its base field

Extension fields with given defining data are unique

(:trac:`20791`)::

sage: K.<a> = NumberField(x^2 + 1)

sage: K.extension(x^2 - 2, 'b') is K.extension(x^2 - 2, 'b')

True

"""

if not isinstance(poly, polynomial_element.Polynomial):

try:

poly = poly.polynomial(self)

except (AttributeError, TypeError):

raise TypeError("polynomial (=%s) must be a polynomial."%repr(poly))

if poly.base_ring() is not self:

poly = poly.change_ring(self)

if names is not None:

name = names

if isinstance(name, tuple):

name = name[0]

return NumberField(poly, name, *args, **kwds)

def factor(self, n):

r"""

Ideal factorization of the principal ideal generated by `n`.

EXAMPLES:

Here we show how to factor Gaussian integers (up to units).

First we form a number field defined by `x^2 + 1`::

sage: K. = NumberField(x^2 + 1); K

Number Field in I with defining polynomial x^2 + 1

Here are the factors::

sage: fi, fj = K.factor(17); fi,fj

((Fractional ideal (I + 4), 1), (Fractional ideal (I - 4), 1))

Now we extract the reduced form of the generators::

sage: zi = fi[0].gens_reduced()[0]; zi

I + 4

sage: zj = fj[0].gens_reduced()[0]; zj

I - 4

We recover the integer that was factored in `\ZZ[i]` (up to a unit)::

sage: zi*zj

-17

One can also factor elements or ideals of the number field::

sage: K.<a> = NumberField(x^2 + 1)

sage: K.factor(1/3)

(Fractional ideal (3))^-1

sage: K.factor(1+a)

Fractional ideal (a + 1)

sage: K.factor(1+a/5)

(Fractional ideal (a + 1)) * (Fractional ideal (-a - 2))^-1 * (Fractional ideal (2*a + 1))^-1 * (Fractional ideal (-3*a - 2))

An example over a relative number field::

sage: pari('setrand(2)')

sage: L. = K.extension(x^2 - 7)

sage: f = L.factor(a + 1); f

(Fractional ideal (1/2*a*b - a + 1/2)) * (Fractional ideal (-1/2*a*b - a + 1/2))

sage: f.value() == a+1

True

It doesn't make sense to factor the ideal (0), so this raises an error::

sage: L.factor(0)

Traceback (most recent call last):

...

AttributeError: 'NumberFieldIdeal' object has no attribute 'factor'

AUTHORS:

- Alex Clemesha (2006-05-20), Francis Clarke (2009-04-21): examples

"""

return self.ideal(n).factor()

def prime_factors(self, x):

"""

Return a list of the prime ideals of self which divide

the ideal generated by `x`.

OUTPUT: list of prime ideals (a new list is returned each time this

function is called)

EXAMPLES::

sage: K.<w> = NumberField(x^2 + 23)

sage: K.prime_factors(w + 1)

[Fractional ideal (2, 1/2*w - 1/2), Fractional ideal (2, 1/2*w + 1/2), Fractional ideal (3, 1/2*w + 1/2)]

"""

return self.ideal(x).prime_factors()

def gen(self, n=0):

"""

Return the generator for this number field.

INPUT:

- ``n`` - must be 0 (the default), or an exception is

raised.

EXAMPLES::

sage: k.<theta> = NumberField(x^14 + 2); k

Number Field in theta with defining polynomial x^14 + 2

sage: k.gen()

theta

sage: k.gen(1)

Traceback (most recent call last):

...

IndexError: Only one generator.

"""

if n != 0:

raise IndexError("Only one generator.")

try:

return self.__gen

except AttributeError:

if self.__polynomial is not None:

X = self.__polynomial.parent().gen()

else:

X = PolynomialRing(QQ).gen()

self.__gen = self._element_class(self, X)

return self.__gen

@cached_method

def _generator_matrix(self):

"""

Return the matrix form of the generator of ``self``.

.. SEEALSO::

:meth:`~sage.rings.number_field.number_field_element.NumberFieldElement.matrix`

EXAMPLES::

sage: x = QQ['x'].gen()

sage: K.<v> = NumberField(x^4 + 514*x^2 + 64321)

sage: R.<r> = NumberField(x^2 + 4*v*x + 5*v^2 + 514)

sage: R._generator_matrix()

[ 0 1]

[-5*v^2 - 514 -4*v]

"""

x = self.gen()

a = x

d = self.relative_degree()

v = x.list()

for n in range(d-1):

a *= x

v += a.list()

from sage.matrix.matrix_space import MatrixSpace

M = MatrixSpace(self.base_ring(), d)

ret = M(v)

ret.set_immutable()

return ret

def is_field(self, proof=True):

"""

Return True since a number field is a field.

EXAMPLES::

sage: NumberField(x^5 + x + 3, 'c').is_field()

True

"""

return True

def is_galois(self):

r"""

Return True if this number field is a Galois extension of

`\QQ`.

EXAMPLES::

sage: NumberField(x^2 + 1, 'i').is_galois()

True

sage: NumberField(x^3 + 2, 'a').is_galois()

False

sage: NumberField(x^15 + x^14 - 14*x^13 - 13*x^12 + 78*x^11 + 66*x^10 - 220*x^9 - 165*x^8 + 330*x^7 + 210*x^6 - 252*x^5 - 126*x^4 + 84*x^3 + 28*x^2 - 8*x - 1, 'a').is_galois()

True

sage: NumberField(x^15 + x^14 - 14*x^13 - 13*x^12 + 78*x^11 + 66*x^10 - 220*x^9 - 165*x^8 + 330*x^7 + 210*x^6 - 252*x^5 - 126*x^4 + 84*x^3 + 28*x^2 - 8*x - 10, 'a').is_galois()

False

"""

#return self.galois_group(type="pari").order() == self.degree()

if self.degree() < 12:

return self.galois_group(type='pari').order() == self.degree()

else:

return len(self.automorphisms()) == self.degree()

@cached_method

def is_abelian(self):

r"""

Return True if this number field is an abelian Galois extension of

`\QQ`.

EXAMPLES::

sage: NumberField(x^2 + 1, 'i').is_abelian()

True

sage: NumberField(x^3 + 2, 'a').is_abelian()

False

sage: NumberField(x^3 + x^2 - 2*x - 1, 'a').is_abelian()

True

sage: NumberField(x^6 + 40*x^3 + 1372, 'a').is_abelian()

False

sage: NumberField(x^6 + x^5 - 5*x^4 - 4*x^3 + 6*x^2 + 3*x - 1, 'a').is_abelian()

True

"""

if not self.is_galois():

return False

d = self.degree()

dsqrt = d.isqrt()

if d == 1 or d.is_prime() or (d == dsqrt**2 and dsqrt.is_prime()):

return True

if d <= 11:

return self.galois_group().is_abelian()

pari_pol = pari(self.polynomial())

return pari_pol.galoisinit().galoisisabelian(1)==1

@cached_method

def galois_group(self, type=None, algorithm='pari', names=None):

r"""

Return the Galois group of the Galois closure of this number field.

INPUT:

- ``type`` - ``none``, ``gap``, or ``pari``. If None (the default),

return an explicit group of automorphisms of self as a

``GaloisGroup_v2`` object. Otherwise, return a ``GaloisGroup_v1``

wrapper object based on a PARI or Gap transitive group object, which

is quicker to compute, but rather less useful (in particular, it

can't be made to act on self). If type = 'gap', the database_gap

package should be installed.

- ``algorithm`` - 'pari', 'kash', 'magma'. (default: 'pari', except

when the degree is >= 12 when 'kash' is tried.)

- ``name`` - a string giving a name for the generator of the Galois

closure of self, when self is not Galois. This is ignored if type is

not None.

Note that computing Galois groups as abstract groups is often much

faster than computing them as explicit automorphism groups (but of

course you get less information out!) For more (important!)

documentation, so the documentation for Galois groups of polynomials

over `\QQ`, e.g., by typing ``K.polynomial().galois_group?``,

where `K` is a number field.

To obtain actual field homomorphisms from the number field to its

splitting field, use type=None.

EXAMPLES:

With type ``None``::

sage: k. = NumberField(x^2 - 14) # a Galois extension

sage: G = k.galois_group(); G

Galois group of Number Field in b with defining polynomial x^2 - 14

sage: G.gen(0)

(1,2)

sage: G.gen(0)(b)

-b

sage: G.artin_symbol(k.primes_above(3)[0])

(1,2)

sage: k. = NumberField(x^3 - x + 1) # not Galois

sage: G = k.galois_group(names='c'); G

Galois group of Galois closure in c of Number Field in b with defining polynomial x^3 - x + 1

sage: G.gen(0)

(1,2,3)(4,5,6)

With type ``'pari'``::

sage: NumberField(x^3-2, 'a').galois_group(type="pari")

Galois group PARI group [6, -1, 2, "S3"] of degree 3 of the Number Field in a with defining polynomial x^3 - 2

sage: NumberField(x-1, 'a').galois_group(type="gap") # optional - database_gap

Galois group Transitive group number 1 of degree 1 of the Number Field in a with defining polynomial x - 1

sage: NumberField(x^2+2, 'a').galois_group(type="gap") # optional - database_gap

Galois group Transitive group number 1 of degree 2 of the Number Field in a with defining polynomial x^2 + 2

sage: NumberField(x^3-2, 'a').galois_group(type="gap") # optional - database_gap

Galois group Transitive group number 2 of degree 3 of the Number Field in a with defining polynomial x^3 - 2

sage: x = polygen(QQ)

sage: NumberField(x^3 + 2*x + 1, 'a').galois_group(type='gap') # optional - database_gap

Galois group Transitive group number 2 of degree 3 of the Number Field in a with defining polynomial x^3 + 2*x + 1

sage: NumberField(x^3 + 2*x + 1, 'a').galois_group(algorithm='magma') # optional - magma database_gap

Galois group Transitive group number 2 of degree 3 of the Number Field in a with defining polynomial x^3 + 2*x + 1

EXPLICIT GALOIS GROUP: We compute the Galois group as an explicit

group of automorphisms of the Galois closure of a field.

sage: K.<a> = NumberField(x^3 - 2)

sage: L.<b1> = K.galois_closure(); L

Number Field in b1 with defining polynomial x^6 + 108

sage: G = End(L); G

Automorphism group of Number Field in b1 with defining polynomial x^6 + 108

sage: G.list()

[

Ring endomorphism of Number Field in b1 with defining polynomial x^6 + 108

Defn: b1 |--> b1,

...

Ring endomorphism of Number Field in b1 with defining polynomial x^6 + 108

Defn: b1 |--> -1/12*b1^4 - 1/2*b1

]

sage: G[2](b1)

1/12*b1^4 + 1/2*b1

"""

from .galois_group import GaloisGroup_v1, GaloisGroup_v2

if type is None:

return GaloisGroup_v2(self, names)

elif type=="pari":

return GaloisGroup_v1(self.absolute_polynomial().galois_group(pari_group=True, algorithm=algorithm), self)

elif type=="gap":

return GaloisGroup_v1(self.absolute_polynomial().galois_group(pari_group=False, algorithm=algorithm), self)

else:

raise ValueError("Galois group type must be None, 'pari', or 'gap'.")

def _normalize_prime_list(self, v):

"""

Internal function to convert into a tuple of primes either None or

a single prime or a list.

EXAMPLES::

sage: K. = NumberField(x^2 + 1)

sage: K._normalize_prime_list(None)

()

sage: K._normalize_prime_list(3)

(3,)

sage: K._normalize_prime_list([3,5])

(3, 5)

"""

if v is None:

v = []

elif not isinstance(v, (list, tuple)):

v = [v]

return tuple(map(ZZ, v))

def power_basis(self):

r"""

Return a power basis for this number field over its base field.

If this number field is represented as `k[t]/f(t)`, then

the basis returned is `1, t, t^2, \ldots, t^{d-1}` where

`d` is the degree of this number field over its base

field.

EXAMPLES::

sage: K.<a> = NumberField(x^5 + 10*x + 1)

sage: K.power_basis()

[1, a, a^2, a^3, a^4]

sage: L. = K.extension(x^2 - 2)

sage: L.power_basis()

[1, b]

sage: L.absolute_field('c').power_basis()

[1, c, c^2, c^3, c^4, c^5, c^6, c^7, c^8, c^9]

sage: M = CyclotomicField(15)

sage: M.power_basis()

[1, zeta15, zeta15^2, zeta15^3, zeta15^4, zeta15^5, zeta15^6, zeta15^7]

"""

g = self.gen()

return [ g**i for i in range(self.relative_degree()) ]

def integral_basis(self, v=None):

"""

Returns a list containing a ZZ-basis for the full ring of integers

of this number field.

INPUT:

- ``v`` - None, a prime, or a list of primes. See the

documentation for self.maximal_order.

EXAMPLES::

sage: K.<a> = NumberField(x^5 + 10*x + 1)

sage: K.integral_basis()

[1, a, a^2, a^3, a^4]

Next we compute the ring of integers of a cubic field in which 2 is

an "essential discriminant divisor", so the ring of integers is not

generated by a single element.

sage: K.<a> = NumberField(x^3 + x^2 - 2*x + 8)

sage: K.integral_basis()

[1, 1/2*a^2 + 1/2*a, a^2]

ALGORITHM: Uses the pari library (via _pari_integral_basis).

"""

return self.maximal_order(v=v).basis()

def _pari_integral_basis(self, v=None, important=True):

"""

Internal function returning an integral basis of this number field in

PARI format.

INPUT:

- ``v`` -- None, a prime, or a list of primes. See the

documentation for self.maximal_order.

- ``important`` -- boolean (default: ``True``). If ``False``,

raise a ``RuntimeError`` if we need to do a difficult

discriminant factorization. This is useful when an integral

basis is not strictly required.

EXAMPLES::

sage: K.<a> = NumberField(x^5 + 10*x + 1)

sage: K._pari_integral_basis()

[1, y, y^2, y^3, y^4]

Next we compute the ring of integers of a cubic field in which 2 is

an "essential discriminant divisor", so the ring of integers is not

generated by a single element.

sage: K.<a> = NumberField(x^3 + x^2 - 2*x + 8)

sage: K._pari_integral_basis()

[1, y, 1/2*y^2 - 1/2*y]

sage: K.integral_basis()

[1, 1/2*a^2 + 1/2*a, a^2]

"""

if (v is None or len(v) == 0) and self._maximize_at_primes:

v = self._maximize_at_primes

v = self._normalize_prime_list(v)

try:

return self._integral_basis_dict[v]

except (AttributeError, KeyError):

f = self.pari_polynomial("y")

if len(v) > 0:

B = f.nfbasis(fa=v)

elif self._assume_disc_small:

B = f.nfbasis(1)

elif not important:

# Trial divide the discriminant with primes up to 10^6

m = self.pari_polynomial().poldisc().abs().factor(limit=10**6)

# Since we only need a *squarefree* factorization for

# primes with exponent 1, we need trial division up to D^(1/3)

# instead of D^(1/2).

trialdivlimit2 = pari(10**12)

trialdivlimit3 = pari(10**18)

if all([ p < trialdivlimit2 or (e == 1 and p < trialdivlimit3) or p.isprime() for p,e in zip(m[0],m[1]) ]):

B = f.nfbasis(fa = m)

else:

raise RuntimeError("Unable to factor discriminant with trial division")

else:

B = f.nfbasis()

self._integral_basis_dict[v] = B

return B

def reduced_basis(self, prec=None):

r"""

This function returns an LLL-reduced basis for the

Minkowski-embedding of the maximal order of a number field.

INPUT:

- ``self`` - number field, the base field

- ``prec (default: None)`` - the precision with which to

compute the Minkowski embedding.

OUTPUT:

An LLL-reduced basis for the Minkowski-embedding of the

maximal order of a number field, given by a sequence of (integral)

elements from the field.

.. note::

In the non-totally-real case, the LLL routine we call is

currently PARI's :pari:`qflll`, which works with floating point

approximations, and so the result is only as good as the

precision promised by PARI. The matrix returned will always

be integral; however, it may only be only "almost" LLL-reduced

when the precision is not sufficiently high.

EXAMPLES::

sage: F.<t> = NumberField(x^6-7*x^4-x^3+11*x^2+x-1)

sage: F.maximal_order().basis()

[1/2*t^5 + 1/2*t^4 + 1/2*t^2 + 1/2, t, t^2, t^3, t^4, t^5]

sage: F.reduced_basis()

[-1, -1/2*t^5 + 1/2*t^4 + 3*t^3 - 3/2*t^2 - 4*t - 1/2, t, 1/2*t^5 + 1/2*t^4 - 4*t^3 - 5/2*t^2 + 7*t + 1/2, 1/2*t^5 - 1/2*t^4 - 2*t^3 + 3/2*t^2 - 1/2, 1/2*t^5 - 1/2*t^4 - 3*t^3 + 5/2*t^2 + 4*t - 5/2]

sage: CyclotomicField(12).reduced_basis()

[1, zeta12^2, zeta12, zeta12^3]

TESTS:

Check that the bug reported at :trac:`10017` is fixed::

sage: x = polygen(QQ)

sage: k.<a> = NumberField(x^6 + 2218926655879913714112*x^4 - 32507675650290949030789018433536*x^3 + 4923635504174417014460581055002374467948544*x^2 - 36066074010564497464129951249279114076897746988630016*x + 264187244046129768986806800244258952598300346857154900812365824)

sage: new_basis = k.reduced_basis(prec=120)

sage: [c.minpoly() for c in new_basis]

[x - 1,

x^2 - x + 1,

x^6 + 3*x^5 - 102*x^4 - 103*x^3 + 10572*x^2 - 59919*x + 127657,

x^6 - 3*x^5 - 102*x^4 + 315*x^3 + 10254*x^2 - 80955*x + 198147,

x^3 - 171*x + 848,

x^6 + 171*x^4 + 1696*x^3 + 29241*x^2 + 145008*x + 719104]

sage: R = k.order(new_basis)

sage: R.discriminant()==k.discriminant()

True

"""

ZK = self.integral_basis()

d = self.absolute_degree()

# If self is totally real, then we can use (x*y).trace() as

# the inner product on the Minkowski embedding, which is

# faster than computing all the conjugates, etc ...

if self.is_totally_real():

from sage.matrix.constructor import matrix

T = pari(matrix(ZZ, d, d, [[(x*y).trace() for x in ZK] for y in ZK])).qflllgram()

else:

M = self.minkowski_embedding(ZK, prec=prec)

T = pari(M).qflll()

return [ sum([ ZZ(T[i][j]) * ZK[j] for j in range(d)]) for i in range(d)]

def reduced_gram_matrix(self, prec=None):

r"""

This function returns the Gram matrix of an LLL-reduced basis for

the Minkowski embedding of the maximal order of a number field.

INPUT:

- ``self`` - number field, the base field

- ``prec (default: None)`` - the precision with which

to calculate the Minkowski embedding. (See NOTE below.)

OUTPUT: The Gram matrix `[\langle x_i,x_j \rangle]` of an LLL reduced

basis for the maximal order of self, where the integral basis for

self is given by `\{x_0, \dots, x_{n-1}\}`. Here `\langle , \rangle` is

the usual inner product on `\RR^n`, and self is embedded in `\RR^n` by

the Minkowski embedding. See the docstring for

:meth:`NumberField_absolute.minkowski_embedding` for more information.

.. note::

In the non-totally-real case, the LLL routine we call is

currently PARI's :pari:`qflll`, which works with floating point

approximations, and so the result is only as good as the

precision promised by PARI. In particular, in this case,

the returned matrix will *not* be integral, and may not

have enough precision to recover the correct gram matrix

(which is known to be integral for theoretical

reasons). Thus the need for the prec flag above.

If the following run-time error occurs: "PariError: not a definite

matrix in lllgram (42)" try increasing the prec parameter,

EXAMPLES::

sage: F.<t> = NumberField(x^6-7*x^4-x^3+11*x^2+x-1)

sage: F.reduced_gram_matrix()

[ 6 3 0 2 0 1]

[ 3 9 0 1 0 -2]

[ 0 0 14 6 -2 3]

[ 2 1 6 16 -3 3]

[ 0 0 -2 -3 16 6]

[ 1 -2 3 3 6 19]

sage: Matrix(6, [(x*y).trace() for x in F.integral_basis() for y in F.integral_basis()])

[2550 133 259 664 1368 3421]

[ 133 14 3 54 30 233]

[ 259 3 54 30 233 217]

[ 664 54 30 233 217 1078]

[1368 30 233 217 1078 1371]

[3421 233 217 1078 1371 5224]

sage: x = polygen(QQ)

sage: F.<alpha> = NumberField(x^4+x^2+712312*x+131001238)

sage: F.reduced_gram_matrix(prec=128)

[ 4.0000000000000000000000000000000000000 0.00000000000000000000000000000000000000 -1.9999999999999999999999999999999999037 -0.99999999999999999999999999999999383702]

[ 0.00000000000000000000000000000000000000 46721.539331563218381658483353092335550 -11488.910026551724275122749703614966768 -418.12718083977141198754424579680468382]

[ -1.9999999999999999999999999999999999037 -11488.910026551724275122749703614966768 5.5658915310500611768713076521847709187e8 1.4179092271494070050433368847682152174e8]

[ -0.99999999999999999999999999999999383702 -418.12718083977141198754424579680468382 1.4179092271494070050433368847682152174e8 1.3665897267919181137884111201405279175e12]

"""

if self.is_totally_real():

try:

return self.__reduced_gram_matrix

except AttributeError:

pass

else:

try:

if self.__reduced_gram_matrix_prec >= prec:

return self.__reduced_gram_matrix

except AttributeError:

pass

from sage.matrix.constructor import matrix

from sage.misc.flatten import flatten

d = self.absolute_degree()

if self.is_totally_real():

B = self.reduced_basis()

self.__reduced_gram_matrix = matrix(ZZ, d, d,

[[(x*y).trace() for x in B]

for y in B])

else:

M = self.minkowski_embedding(prec=prec)

T = matrix(d, flatten([ a.vector().list()

for a in self.reduced_basis(prec=prec) ]))

A = M*(T.transpose())

self.__reduced_gram_matrix = A.transpose()*A

if prec is None:

## this is the default choice for minkowski_embedding

self.__reduced_gram_matrix_prec = 53

else:

self.__reduced_gram_matrix_prec = prec

return self.__reduced_gram_matrix

#******************************************************

# Supplementary algorithm to enumerate lattice points

#******************************************************

def _positive_integral_elements_with_trace(self, C):

r"""

Find all totally positive integral elements in self whose

trace is between C[0] and C[1], inclusive.

.. note::

This is currently only implemented in the case that self is

totally real, since it requires exact computation of

:meth:`.reduced_gram_matrix`.

EXAMPLES::

sage: K.<alpha> = NumberField(ZZ['x'].0^2-2)

sage: K._positive_integral_elements_with_trace([0,5])

[alpha + 2, -alpha + 2, 2, 1]

sage: L.<beta> = NumberField(ZZ['x'].0^2+1)

sage: L._positive_integral_elements_with_trace([5,11])

Traceback (most recent call last):

...

NotImplementedError: exact computation of LLL reduction only implemented in the totally real case

sage: L._positive_integral_elements_with_trace([-5,1])

Traceback (most recent call last):

...

ValueError: bounds must be positive

"""

if C[0] < 0:

raise ValueError("bounds must be positive")

if not self.is_totally_real():

raise NotImplementedError("exact computation of LLL reduction only implemented in the totally real case")

Z_F = self.maximal_order()

B = self.reduced_basis()

T = self.reduced_gram_matrix()

P = pari(T).qfminim((C[1]**2)*(1./2), 10**6)[2]

S = []

for p in P:

theta = sum([ p.list()[i]*B[i] for i in range(self.degree())])

if theta.trace() < 0:

theta *= -1

if theta.trace() >= C[0] and theta.trace() <= C[1]:

if self(theta).is_totally_positive():

S.append(self(theta))

return S

def zeta_function(self, prec=53,

max_imaginary_part=0,

max_asymp_coeffs=40):

r"""

Return the Zeta function of this number field.

This actually returns an interface to Tim Dokchitser's program for

computing with the Dedekind zeta function zeta_F(s) of the number

field F.

INPUT:

- ``prec`` - integer (bits precision)

- ``max_imaginary_part`` - real number

- ``max_asymp_coeffs`` - integer

OUTPUT: The zeta function of this number field.

EXAMPLES::

sage: K.<a> = NumberField(ZZ['x'].0^2+ZZ['x'].0-1)

sage: Z = K.zeta_function()

sage: Z

Zeta function associated to Number Field in a with defining polynomial x^2 + x - 1

sage: Z(-1)

0.0333333333333333

sage: L.<a, b, c> = NumberField([x^2 - 5, x^2 + 3, x^2 + 1])

sage: Z = L.zeta_function()

sage: Z(5)

1.00199015670185

"""

from sage.lfunctions.all import Dokchitser

key = (prec, max_imaginary_part, max_asymp_coeffs)

r1 = self.signature()[0]

r2 = self.signature()[1]

zero = [0]

one = [1]

Z = Dokchitser(conductor = abs(self.absolute_discriminant()),

gammaV = (r1+r2)*zero + r2*one,

weight = 1,

eps = 1,

poles = [1],

prec = prec)

s = 'nf = nfinit(%s);'%self.absolute_polynomial()

s += 'dzk = dirzetak(nf,cflength());'

Z.init_coeffs('dzk[k]', pari_precode=s,

max_imaginary_part=max_imaginary_part,

max_asymp_coeffs=max_asymp_coeffs)

Z.check_functional_equation()

Z.rename('Zeta function associated to %s'%self)

return Z

@cached_method

def narrow_class_group(self, proof=None):

r"""

Return the narrow class group of this field.

INPUT:

- ``proof`` - default: None (use the global proof

setting, which defaults to True).

EXAMPLES::

sage: NumberField(x^3+x+9, 'a').narrow_class_group()

Multiplicative Abelian group isomorphic to C2

TESTS::

sage: QuadraticField(3, 'a').narrow_class_group()

Multiplicative Abelian group isomorphic to C2

"""

proof = proof_flag(proof)

k = self.pari_bnf(proof)

s = k.bnfnarrow().sage()

return sage.groups.abelian_gps.abelian_group.AbelianGroup(s[1])

def ngens(self):

"""

Return the number of generators of this number field (always 1).

OUTPUT: the python integer 1.

EXAMPLES::

sage: NumberField(x^2 + 17,'a').ngens()

sage: NumberField(x + 3,'a').ngens()

sage: k.<a> = NumberField(x + 3)

sage: k.ngens()

sage: k.0

-3

"""

return 1

def order(self):

"""

Return the order of this number field (always +infinity).

OUTPUT: always positive infinity

EXAMPLES::

sage: NumberField(x^2 + 19,'a').order()

+Infinity

"""

return infinity.infinity

def absolute_polynomial_ntl(self):

r"""

Alias for :meth:`~polynomial_ntl`. Mostly for internal use.

EXAMPLES::

sage: NumberField(x^2 + (2/3)*x - 9/17,'a').absolute_polynomial_ntl()

([-27 34 51], 51)

"""

return self.polynomial_ntl()

def polynomial_ntl(self):

"""

Return defining polynomial of this number field as a pair, an ntl

polynomial and a denominator.

This is used mainly to implement some internal arithmetic.

EXAMPLES::

sage: NumberField(x^2 + (2/3)*x - 9/17,'a').polynomial_ntl()

([-27 34 51], 51)

"""

try:

return (self.__polynomial_ntl, self.__denominator_ntl)

except AttributeError:

self.__denominator_ntl = ntl.ZZ()

den = self.polynomial().denominator()

self.__denominator_ntl.set_from_sage_int(ZZ(den))

self.__polynomial_ntl = ntl.ZZX((self.polynomial()*den).list())

return (self.__polynomial_ntl, self.__denominator_ntl)

def polynomial(self):

"""

Return the defining polynomial of this number field.

This is exactly the same as

``self.defining_polynomial()``.

EXAMPLES::

sage: NumberField(x^2 + (2/3)*x - 9/17,'a').polynomial()

x^2 + 2/3*x - 9/17

"""

return self.__polynomial

def defining_polynomial(self): # do not overload this -- overload polynomial instead

"""

Return the defining polynomial of this number field.

This is exactly the same as ``self.polynomial()``.

EXAMPLES::

sage: k5.<z> = CyclotomicField(5)

sage: k5.defining_polynomial()

x^4 + x^3 + x^2 + x + 1

sage: y = polygen(QQ,'y')

sage: k.<a> = NumberField(y^9 - 3*y + 5); k

Number Field in a with defining polynomial y^9 - 3*y + 5

sage: k.defining_polynomial()

y^9 - 3*y + 5

"""

return self.polynomial()

def polynomial_ring(self):

"""

Return the polynomial ring that we view this number field as being

a quotient of (by a principal ideal).

EXAMPLES: An example with an absolute field::

sage: k.<a> = NumberField(x^2 + 3)

sage: y = polygen(QQ, 'y')

sage: k.<a> = NumberField(y^2 + 3)

sage: k.polynomial_ring()

Univariate Polynomial Ring in y over Rational Field

An example with a relative field::

sage: y = polygen(QQ, 'y')

sage: M.<a> = NumberField([y^3 + 97, y^2 + 1]); M

Number Field in a0 with defining polynomial y^3 + 97 over its base field

sage: M.polynomial_ring()

Univariate Polynomial Ring in y over Number Field in a1 with defining polynomial y^2 + 1

"""

return self.relative_polynomial().parent()

def polynomial_quotient_ring(self):

"""

Return the polynomial quotient ring isomorphic to this number

field.

EXAMPLES::

sage: K = NumberField(x^3 + 2*x - 5, 'alpha')

sage: K.polynomial_quotient_ring()

Univariate Quotient Polynomial Ring in alpha over Rational Field with modulus x^3 + 2*x - 5

"""

return self.polynomial_ring().quotient(self.relative_polynomial(), self.variable_name())

def regulator(self, proof=None):

"""

Return the regulator of this number field.

Note that PARI computes the regulator to higher precision than the

Sage default.

INPUT:

- ``proof`` - default: True, unless you set it

otherwise.

EXAMPLES::

sage: NumberField(x^2-2, 'a').regulator()

0.881373587019543

sage: NumberField(x^4+x^3+x^2+x+1, 'a').regulator()

0.962423650119207

"""

proof = proof_flag(proof)

try:

return self.__regulator

except AttributeError:

from sage.rings.all import RealField

k = self.pari_bnf(proof)

self.__regulator = RealField(53)(k.bnf_get_reg())

return self.__regulator

def residue_field(self, prime, names=None, check=True):

"""

Return the residue field of this number field at a given prime, ie

`O_K / p O_K`.

INPUT:

- ``prime`` - a prime ideal of the maximal order in

this number field, or an element of the field which generates a

principal prime ideal.

- ``names`` - the name of the variable in the residue

field

- ``check`` - whether or not to check the primality of

prime.

OUTPUT: The residue field at this prime.

EXAMPLES::

sage: R.<x> = QQ[]

sage: K.<a> = NumberField(x^4+3*x^2-17)

sage: P = K.ideal(61).factor()[0][0]

sage: K.residue_field(P)

Residue field in abar of Fractional ideal (61, a^2 + 30)

sage: K. = NumberField(x^2 + 1)

sage: K.residue_field(1+i)

Residue field of Fractional ideal (i + 1)

TESTS::

sage: L. = NumberField(x^2 + 5)

sage: L.residue_field(P)

Traceback (most recent call last):

...

ValueError: Fractional ideal (61, a^2 + 30) is not an ideal of Number Field in b with defining polynomial x^2 + 5

sage: L.residue_field(2)

Traceback (most recent call last):

...

ValueError: Fractional ideal (2) is not a prime ideal

sage: L.residue_field(L.prime_above(5)^2)

Traceback (most recent call last):

...

ValueError: Fractional ideal (5) is not a prime ideal

"""

from sage.rings.number_field.number_field_ideal import is_NumberFieldIdeal

if is_NumberFieldIdeal(prime) and prime.number_field() is not self:

raise ValueError("%s is not an ideal of %s"%(prime,self))

# This allows principal ideals to be specified using a generator:

try:

prime = self.ideal(prime)

except TypeError:

pass

if not is_NumberFieldIdeal(prime) or prime.number_field() is not self:

raise ValueError("%s is not an ideal of %s"%(prime,self))

if check and not prime.is_prime():

raise ValueError("%s is not a prime ideal"%prime)

from sage.rings.finite_rings.residue_field import ResidueField

return ResidueField(prime, names=names, check=False)

def signature(self):

"""

Return (r1, r2), where r1 and r2 are the number of real embeddings

and pairs of complex embeddings of this field, respectively.

EXAMPLES::

sage: NumberField(x^2+1, 'a').signature()

(0, 1)

sage: NumberField(x^3-2, 'a').signature()

(1, 1)

"""

r1, r2 = self.pari_nf().nf_get_sign()

return (ZZ(r1), ZZ(r2))

def trace_pairing(self, v):

"""

Return the matrix of the trace pairing on the elements of the list

`v`.

EXAMPLES::

sage: K.<zeta3> = NumberField(x^2 + 3)

sage: K.trace_pairing([1,zeta3])

[ 2 0]

[ 0 -6]

"""

import sage.matrix.matrix_space

A = sage.matrix.matrix_space.MatrixSpace(self.base_ring(), len(v))(0)

for i in range(len(v)):

for j in range(i,len(v)):

t = (self(v[i]*v[j])).trace()

A[i,j] = t

A[j,i] = t

return A

def uniformizer(self, P, others="positive"):

"""

Returns an element of self with valuation 1 at the prime ideal P.

INPUT:

- ``self`` - a number field

- ``P`` - a prime ideal of self

- ``others`` - either "positive" (default), in which

case the element will have non-negative valuation at all other

primes of self, or "negative", in which case the element will have

non-positive valuation at all other primes of self.

.. note::

When P is principal (e.g. always when self has class number

one) the result may or may not be a generator of P!

EXAMPLES::

sage: K.<a> = NumberField(x^2 + 5); K

Number Field in a with defining polynomial x^2 + 5

sage: P,Q = K.ideal(3).prime_factors()

sage: P

Fractional ideal (3, a + 1)

sage: pi = K.uniformizer(P); pi

a + 1

sage: K.ideal(pi).factor()

(Fractional ideal (2, a + 1)) * (Fractional ideal (3, a + 1))

sage: pi = K.uniformizer(P,'negative'); pi

1/2*a + 1/2

sage: K.ideal(pi).factor()

(Fractional ideal (2, a + 1))^-1 * (Fractional ideal (3, a + 1))

sage: K = CyclotomicField(9)

sage: Plist=K.ideal(17).prime_factors()

sage: pilist = [K.uniformizer(P) for P in Plist]

sage: [pi.is_integral() for pi in pilist]

[True, True, True]

sage: [pi.valuation(P) for pi,P in zip(pilist,Plist)]

[1, 1, 1]

sage: [ pilist[i] in Plist[i] for i in range(len(Plist)) ]

[True, True, True]

sage: K.<t> = NumberField(x^4 - x^3 - 3*x^2 - x + 1)

sage: [K.uniformizer(P) for P,e in factor(K.ideal(2))]

[2]

sage: [K.uniformizer(P) for P,e in factor(K.ideal(3))]

[t - 1]

sage: [K.uniformizer(P) for P,e in factor(K.ideal(5))]

[t^2 - t + 1, t + 2, t - 2]

sage: [K.uniformizer(P) for P,e in factor(K.ideal(7))]

[t^2 + 3*t + 1]

sage: [K.uniformizer(P) for P,e in factor(K.ideal(67))]

[t + 23, t + 26, t - 32, t - 18]

ALGORITHM:

Use PARI. More precisely, use the second component of

:pari:`idealprimedec` in the "positive" case. Use :pari:`idealappr`

with exponent of -1 and invert the result in the "negative"

case.

"""

if not is_NumberFieldIdeal(P):

P = self.ideal(P)

P = P.pari_prime()

if others == "positive":

return self(P[1])

elif others == "negative":

nf = self.pari_nf()

F = pari.matrix(1, 2, [P, -1])

return ~self(nf.idealappr(F, 1))

else:

raise ValueError("others must be 'positive' or 'negative'")

def units(self, proof=None):

"""

Return generators for the unit group modulo torsion.

ALGORITHM: Uses PARI's :pari:`bnfunit` command.

INPUT:

- ``proof`` (bool, default True) flag passed to ``pari``.

.. note::

For more functionality see the unit_group() function.

.. SEEALSO::

:meth:`unit_group`

:meth:`S_unit_group`

:meth:`S_units`

EXAMPLES::

sage: x = polygen(QQ)

sage: A = x^4 - 10*x^3 + 20*5*x^2 - 15*5^2*x + 11*5^3

sage: K = NumberField(A, 'a')

sage: K.units()

(1/275*a^3 - 7/55*a^2 + 6/11*a - 3,)

For big number fields, provably computing the unit group can

take a very long time. In this case, one can ask for the

conjectural unit group (correct if the Generalized Riemann

Hypothesis is true)::

sage: K = NumberField(x^17 + 3, 'a')

sage: K.units(proof=True) # takes forever, not tested

...

sage: K.units(proof=False) # result not independently verified

(a^9 + a - 1,

a^15 - a^12 + a^10 - a^9 - 2*a^8 + 3*a^7 + a^6 - 3*a^5 + a^4 + 4*a^3 - 3*a^2 - 2*a + 2,

a^16 - a^15 + a^14 - a^12 + a^11 - a^10 - a^8 + a^7 - 2*a^6 + a^4 - 3*a^3 + 2*a^2 - 2*a + 1,

2*a^16 - a^14 - a^13 + 3*a^12 - 2*a^10 + a^9 + 3*a^8 - 3*a^6 + 3*a^5 + 3*a^4 - 2*a^3 - 2*a^2 + 3*a + 4,

2*a^16 - 3*a^15 + 3*a^14 - 3*a^13 + 3*a^12 - a^11 + a^9 - 3*a^8 + 4*a^7 - 5*a^6 + 6*a^5 - 4*a^4 + 3*a^3 - 2*a^2 - 2*a + 4,

a^16 - a^15 - 3*a^14 - 4*a^13 - 4*a^12 - 3*a^11 - a^10 + 2*a^9 + 4*a^8 + 5*a^7 + 4*a^6 + 2*a^5 - 2*a^4 - 6*a^3 - 9*a^2 - 9*a - 7,

a^15 + a^14 + 2*a^11 + a^10 - a^9 + a^8 + 2*a^7 - a^5 + 2*a^3 - a^2 - 3*a + 1,

5*a^16 - 6*a^14 + a^13 + 7*a^12 - 2*a^11 - 7*a^10 + 4*a^9 + 7*a^8 - 6*a^7 - 7*a^6 + 8*a^5 + 6*a^4 - 11*a^3 - 5*a^2 + 13*a + 4)

TESTS:

Number fields defined by non-monic and non-integral

polynomials are supported (:trac:`252`)::

sage: K.<a> = NumberField(1/2*x^2 - 1/6)

sage: K.units()

(3*a - 2,)

"""

proof = proof_flag(proof)

# if we have cached provable results, return them immediately

try:

return self.__units

except AttributeError:

pass

# if proof==False and we have cached results, return them immediately

if not proof:

try:

return self.__units_no_proof

except AttributeError:

pass

# get PARI to compute the units

B = self.pari_bnf(proof).bnfunit()

B = tuple(self(b, check=False) for b in B)

if proof:

# cache the provable results and return them

self.__units = B

return self.__units

else:

# cache the conjectural results and return them

self.__units_no_proof = B

return self.__units_no_proof

def unit_group(self, proof=None):

"""

Return the unit group (including torsion) of this number field.

ALGORITHM: Uses PARI's :pari:`bnfunit` command.

INPUT:

- ``proof`` (bool, default True) flag passed to ``pari``.

.. note::

The group is cached.

.. SEEALSO::

:meth:`units`

:meth:`S_unit_group`

:meth:`S_units`

EXAMPLES::

sage: x = QQ['x'].0

sage: A = x^4 - 10*x^3 + 20*5*x^2 - 15*5^2*x + 11*5^3

sage: K = NumberField(A, 'a')

sage: U = K.unit_group(); U

Unit group with structure C10 x Z of Number Field in a with defining polynomial x^4 - 10*x^3 + 100*x^2 - 375*x + 1375

sage: U.gens()

(u0, u1)

sage: U.gens_values() # random

[-1/275*a^3 + 7/55*a^2 - 6/11*a + 4, 1/275*a^3 + 4/55*a^2 - 5/11*a + 3]

sage: U.invariants()

(10, 0)

sage: [u.multiplicative_order() for u in U.gens()]

[10, +Infinity]

For big number fields, provably computing the unit group can

take a very long time. In this case, one can ask for the

conjectural unit group (correct if the Generalized Riemann

Hypothesis is true)::

sage: K = NumberField(x^17 + 3, 'a')

sage: K.unit_group(proof=True) # takes forever, not tested

...

sage: U = K.unit_group(proof=False)

sage: U

Unit group with structure C2 x Z x Z x Z x Z x Z x Z x Z x Z of Number Field in a with defining polynomial x^17 + 3

sage: U.gens()

(u0, u1, u2, u3, u4, u5, u6, u7, u8)

sage: U.gens_values() # result not independently verified

[-1, a^9 + a - 1, a^15 - a^12 + a^10 - a^9 - 2*a^8 + 3*a^7 + a^6 - 3*a^5 + a^4 + 4*a^3 - 3*a^2 - 2*a + 2, a^16 - a^15 + a^14 - a^12 + a^11 - a^10 - a^8 + a^7 - 2*a^6 + a^4 - 3*a^3 + 2*a^2 - 2*a + 1, 2*a^16 - a^14 - a^13 + 3*a^12 - 2*a^10 + a^9 + 3*a^8 - 3*a^6 + 3*a^5 + 3*a^4 - 2*a^3 - 2*a^2 + 3*a + 4, 2*a^16 - 3*a^15 + 3*a^14 - 3*a^13 + 3*a^12 - a^11 + a^9 - 3*a^8 + 4*a^7 - 5*a^6 + 6*a^5 - 4*a^4 + 3*a^3 - 2*a^2 - 2*a + 4, a^16 - a^15 - 3*a^14 - 4*a^13 - 4*a^12 - 3*a^11 - a^10 + 2*a^9 + 4*a^8 + 5*a^7 + 4*a^6 + 2*a^5 - 2*a^4 - 6*a^3 - 9*a^2 - 9*a - 7, a^15 + a^14 + 2*a^11 + a^10 - a^9 + a^8 + 2*a^7 - a^5 + 2*a^3 - a^2 - 3*a + 1, 5*a^16 - 6*a^14 + a^13 + 7*a^12 - 2*a^11 - 7*a^10 + 4*a^9 + 7*a^8 - 6*a^7 - 7*a^6 + 8*a^5 + 6*a^4 - 11*a^3 - 5*a^2 + 13*a + 4]

"""

proof = proof_flag(proof)

try:

return self._unit_group

except AttributeError:

pass

if not proof:

try:

return self._unit_group_no_proof

except AttributeError:

pass

U = UnitGroup(self,proof)

if proof:

self._unit_group = U

else:

self._unit_group_no_proof = U

return U

def S_unit_group(self, proof=None, S=None):

"""

Return the S-unit group (including torsion) of this number field.

ALGORITHM: Uses PARI's :pari:`bnfsunit` command.

INPUT:

- ``proof`` (bool, default True) flag passed to ``pari``.

- ``S`` - list or tuple of prime ideals, or an ideal, or a single

ideal or element from which an ideal can be constructed, in

which case the support is used. If None, the global unit

group is constructed; otherwise, the S-unit group is

constructed.

.. note::

The group is cached.

EXAMPLES::

sage: x = polygen(QQ)

sage: K.<a> = NumberField(x^4 - 10*x^3 + 20*5*x^2 - 15*5^2*x + 11*5^3)

sage: U = K.S_unit_group(S=a); U

S-unit group with structure C10 x Z x Z x Z of Number Field in a with defining polynomial x^4 - 10*x^3 + 100*x^2 - 375*x + 1375 with S = (Fractional ideal (5, 1/275*a^3 + 4/55*a^2 - 5/11*a + 5), Fractional ideal (11, 1/275*a^3 + 4/55*a^2 - 5/11*a + 9))

sage: U.gens()

(u0, u1, u2, u3)

sage: U.gens_values() # random

[-1/275*a^3 + 7/55*a^2 - 6/11*a + 4, 1/275*a^3 + 4/55*a^2 - 5/11*a + 3, 1/275*a^3 + 4/55*a^2 - 5/11*a + 5, -14/275*a^3 + 21/55*a^2 - 29/11*a + 6]

sage: U.invariants()

(10, 0, 0, 0)

sage: [u.multiplicative_order() for u in U.gens()]

[10, +Infinity, +Infinity, +Infinity]

sage: U.primes()

(Fractional ideal (5, 1/275*a^3 + 4/55*a^2 - 5/11*a + 5), Fractional ideal (11, 1/275*a^3 + 4/55*a^2 - 5/11*a + 9))

With the default value of `S`, the S-unit group is the same as

the global unit group::

sage: x = polygen(QQ)

sage: K.<a> = NumberField(x^3 + 3)

sage: U = K.unit_group(proof=False)

sage: U == K.S_unit_group(proof=False)

True

The value of `S` may be specified as a list of prime ideals,

or an ideal, or an element of the field::

sage: K.<a> = NumberField(x^3 + 3)

sage: U = K.S_unit_group(proof=False, S=K.ideal(6).prime_factors()); U

S-unit group with structure C2 x Z x Z x Z x Z of Number Field in a with defining polynomial x^3 + 3 with S = (Fractional ideal (-a^2 + a - 1), Fractional ideal (a + 1), Fractional ideal (a))

sage: K.<a> = NumberField(x^3 + 3)

sage: U = K.S_unit_group(proof=False, S=K.ideal(6)); U

S-unit group with structure C2 x Z x Z x Z x Z of Number Field in a with defining polynomial x^3 + 3 with S = (Fractional ideal (-a^2 + a - 1), Fractional ideal (a + 1), Fractional ideal (a))

sage: K.<a> = NumberField(x^3 + 3)

sage: U = K.S_unit_group(proof=False, S=6); U

S-unit group with structure C2 x Z x Z x Z x Z of Number Field in a with defining polynomial x^3 + 3 with S = (Fractional ideal (-a^2 + a - 1), Fractional ideal (a + 1), Fractional ideal (a))

sage: U

S-unit group with structure C2 x Z x Z x Z x Z of Number Field in a with defining polynomial x^3 + 3 with S = (Fractional ideal (-a^2 + a - 1), Fractional ideal (a + 1), Fractional ideal (a))

sage: U.primes()

(Fractional ideal (-a^2 + a - 1),

Fractional ideal (a + 1),

Fractional ideal (a))

sage: U.gens()

(u0, u1, u2, u3, u4)

sage: U.gens_values()

[-1, a^2 - 2, -a^2 + a - 1, a + 1, a]

The exp and log methods can be used to create `S`-units from

sequences of exponents, and recover the exponents::

sage: U.gens_orders()

(2, 0, 0, 0, 0)

sage: u = U.exp((3,1,4,1,5)); u

-6*a^2 + 18*a - 54

sage: u.norm().factor()

-1 * 2^9 * 3^5

sage: U.log(u)

(1, 1, 4, 1, 5)

"""

proof = proof_flag(proof)

# process the parameter S:

if not S:

S = ()

else:

if isinstance(S, list):

S = tuple(S)

if not isinstance(S, tuple):

try:

S = tuple(self.ideal(S).prime_factors())

except (NameError, TypeError, ValueError):

raise ValueError("Cannot make a set of primes from %s"%(S,))

else:

try:

S = tuple(self.ideal(P) for P in S)

except (NameError, TypeError, ValueError):

raise ValueError("Cannot make a set of primes from %s"%(S,))

if not all([P.is_prime() for P in S]):

raise ValueError("Not all elements of %s are prime ideals"%(S,))

try:

return self._S_unit_group_cache[S]

except AttributeError:

self._S_unit_group_cache = {}

except KeyError:

pass

if not proof:

try:

return self._S_unit_group_no_proof_cache[S]

except AttributeError:

self._S_unit_group_no_proof_cache = {}

except KeyError:

pass

U = UnitGroup(self,proof,S=S)

if proof:

self._S_unit_group_cache[S] = U

else:

self._S_unit_group_no_proof_cache[S] = U

return U

def zeta(self, n=2, all=False):

"""

Return one, or a list of all, primitive n-th root of unity in this field.

INPUT:

- ``n`` - positive integer

- ``all`` - bool. If False (default), return a primitive

`n`-th root of unity in this field, or raise a ValueError

exception if there are none. If True, return a list of

all primitive `n`-th roots of unity in this field

(possibly empty).

.. note::

To obtain the maximal order of a root of unity in this field,

use self.number_of_roots_of_unity().

.. note::

We do not create the full unit group since that can be

expensive, but we do use it if it is already known.

EXAMPLES::

sage: K.<z> = NumberField(x^2 + 3)

sage: K.zeta(1)

sage: K.zeta(2)

-1

sage: K.zeta(2, all=True)

[-1]

sage: K.zeta(3)

1/2*z - 1/2

sage: K.zeta(3, all=True)

[1/2*z - 1/2, -1/2*z - 1/2]

sage: K.zeta(4)

Traceback (most recent call last):

...

ValueError: There are no 4th roots of unity in self.

sage: r.<x> = QQ[]

sage: K. = NumberField(x^2+1)

sage: K.zeta(4)

sage: K.zeta(4,all=True)

[b, -b]

sage: K.zeta(3)

Traceback (most recent call last):

...

ValueError: There are no 3rd roots of unity in self.

sage: K.zeta(3,all=True)

[]

Number fields defined by non-monic and non-integral

polynomials are supported (:trac:`252`)::

sage: K.<a> = NumberField(1/2*x^2 + 1/6)

sage: K.zeta(3)

-3/2*a - 1/2

"""

try:

return self._unit_group.zeta(n,all)

except AttributeError:

pass

try:

return self._unit_group_no_proof.zeta(n,all)

except AttributeError:

pass

K = self

n = ZZ(n)

if n <= 0:

raise ValueError("n (=%s) must be positive"%n)

if n == 1:

if all:

return [K(1)]

else:

return K(1)

elif n == 2:

if all:

return [K(-1)]

else:

return K(-1)

# First check if the degree of K is compatible with an

# inclusion QQ(\zeta_n) -> K.

if sage.arith.all.euler_phi(n).divides(K.absolute_degree()):

# Factor the n-th cyclotomic polynomial over K.

f = K.pari_polynomial('y')

factors = f.nffactor(pari.polcyclo(n)).component(1)

roots = [K(-g.polcoeff(0)) for g in factors if g.poldegree() == 1]

if all:

return roots

if roots:

return roots[0]

raise ValueError("There are no %s roots of unity in self." % n.ordinal_str())

def zeta_order(self):

r"""

Return the number of roots of unity in this field.

.. note::

We do not create the full unit group since that can be

expensive, but we do use it if it is already known.

EXAMPLES::

sage: F.<alpha> = NumberField(x**22+3)

sage: F.zeta_order()

sage: F.<alpha> = NumberField(x**2-7)

sage: F.zeta_order()

TESTS:

Number fields defined by non-monic and non-integral

polynomials are supported (:trac:`252`)::

sage: K.<a> = NumberField(1/2*x^2 + 1/6)

sage: K.zeta_order()

"""

try:

return self._unit_group.zeta_order()

except AttributeError:

pass

try:

return self._unit_group_no_proof.zeta_order()

except AttributeError:

pass

return ZZ(self.pari_nf().nfrootsof1()[0])

number_of_roots_of_unity = zeta_order

def primitive_root_of_unity(self):

"""

Return a generator of the roots of unity in this field.

OUTPUT: a primitive root of unity. No guarantee is made about

which primitive root of unity this returns, not even for

cyclotomic fields. Repeated calls of this function may return

a different value.

.. note::

We do not create the full unit group since that can be

expensive, but we do use it if it is already known.

EXAMPLES::

sage: K. = NumberField(x^2+1)

sage: z = K.primitive_root_of_unity(); z

sage: z.multiplicative_order()

sage: K.<a> = NumberField(x^2+x+1)

sage: z = K.primitive_root_of_unity(); z

a + 1

sage: z.multiplicative_order()

sage: x = polygen(QQ)

sage: F.<a,b> = NumberField([x^2 - 2, x^2 - 3])

sage: y = polygen(F)

sage: K.<c> = F.extension(y^2 - (1 + a)*(a + b)*a*b)

sage: K.primitive_root_of_unity()

-1

We do not special-case cyclotomic fields, so we do not always

get the most obvious primitive root of unity::

sage: K.<a> = CyclotomicField(3)

sage: z = K.primitive_root_of_unity(); z

a + 1

sage: z.multiplicative_order()

sage: K = CyclotomicField(3)

sage: z = K.primitive_root_of_unity(); z

zeta3 + 1

sage: z.multiplicative_order()

TESTS:

Check for :trac:`15027`. We use a new variable name::

sage: K.<f> = NumberField(x^2 + x + 1)

sage: K.primitive_root_of_unity()

f + 1

sage: UK = K.unit_group()

sage: K.primitive_root_of_unity()

f + 1

Number fields defined by non-monic and non-integral

polynomials are supported (:trac:`252`)::

sage: K.<a> = NumberField(3*x^2 + 1)

sage: K.primitive_root_of_unity()

-3/2*a + 1/2

"""

try:

return self._unit_group.torsion_generator().value()

except AttributeError:

pass

try:

return self._unit_group_no_proof.torsion_generator().value()

except AttributeError:

pass

pK = self.pari_nf()

n, z = pK.nfrootsof1()

return self(z, check=False)

def roots_of_unity(self):

"""

Return all the roots of unity in this field, primitive or not.

EXAMPLES::

sage: K. = NumberField(x^2+1)

sage: zs = K.roots_of_unity(); zs

[b, -1, -b, 1]

sage: [ z**K.number_of_roots_of_unity() for z in zs ]

[1, 1, 1, 1]

"""

z = self.primitive_root_of_unity()

n = self.zeta_order()

return [ z**k for k in range(1, n+1) ]

def zeta_coefficients(self, n):

"""

Compute the first n coefficients of the Dedekind zeta function of

this field as a Dirichlet series.

EXAMPLES::

sage: x = QQ['x'].0

sage: NumberField(x^2+1, 'a').zeta_coefficients(10)

[1, 1, 0, 1, 2, 0, 0, 1, 1, 2]

"""

return self.pari_nf().dirzetak(n)

def solve_CRT(self, reslist, Ilist, check=True):

r"""

Solve a Chinese remainder problem over this number field.

INPUT:

- ``reslist`` -- a list of residues, i.e. integral number field elements

- ``Ilist`` -- a list of integral ideals, assumed pairwise coprime

- ``check`` (boolean, default True) -- if True, result is checked

OUTPUT:

An integral element x such that x-reslist[i] is in Ilist[i] for all i.

.. note::

The current implementation requires the ideals to be pairwise

coprime. A more general version would be possible.

EXAMPLES::

sage: K.<a> = NumberField(x^2-10)

sage: Ilist = [K.primes_above(p)[0] for p in prime_range(10)]

sage: b = K.solve_CRT([1,2,3,4],Ilist,True)

sage: all([b-i-1 in Ilist[i] for i in range(4)])

True

sage: Ilist = [K.ideal(a), K.ideal(2)]

sage: K.solve_CRT([0,1],Ilist,True)

Traceback (most recent call last):

...

ArithmeticError: ideals in solve_CRT() must be pairwise coprime

sage: Ilist[0]+Ilist[1]

Fractional ideal (2, a)

"""

n = len(reslist)

try:

reslist = [self(x) for x in reslist]

except ValueError:

raise ValueError("solve_CRT requires a list of arguments in the field")

if n==0:

return self.zero()

if n==1:

return reslist[0]

if n==2:

try:

r = Ilist[0].element_1_mod(Ilist[1])

except TypeError:

raise ArithmeticError("ideals in solve_CRT() must be pairwise coprime")

x = ((1-r)*reslist[0]+r*reslist[1]).mod(prod(Ilist))

else: # n>2;, use induction / recursion

x = self.solve_CRT([reslist[0],self.solve_CRT(reslist[1:],Ilist[1:])],

[Ilist[0],prod(Ilist[1:])], check=check)

if check and not all([x-xi in Ii for xi,Ii in zip(reslist, Ilist)]):

raise RuntimeError("Error in number field solve_CRT()")

return self(x)

def valuation(self, prime):

r"""

Return the valuation on this field defined by ``prime``.

INPUT:

- ``prime`` -- a prime that does not split, a discrete

(pseudo-)valuation or a fractional ideal

EXAMPLES:

The valuation can be specified with an integer ``prime`` that is

completely ramified in ``R``::

sage: K.<a> = NumberField(x^2 + 1)

sage: K.valuation(2)

2-adic valuation

It can also be unramified in ``R``::

sage: K.valuation(3)

3-adic valuation

A ``prime`` that factors into pairwise distinct factors, results in an error::

sage: K.valuation(5)

Traceback (most recent call last):

...

ValueError: The valuation Gauss valuation induced by 5-adic valuation does not approximate a unique extension of 5-adic valuation with respect to x^2 + 1

The valuation can also be selected by giving a valuation on the base

ring that extends uniquely::

sage: CyclotomicField(5).valuation(ZZ.valuation(5))

5-adic valuation

When the extension is not unique, this does not work::

sage: K.valuation(ZZ.valuation(5))

Traceback (most recent call last):

...

ValueError: The valuation Gauss valuation induced by 5-adic valuation does not approximate a unique extension of 5-adic valuation with respect to x^2 + 1

For a number field which is of the form `K[x]/(G)`, you can specify a

valuation by providing a discrete pseudo-valuation on `K[x]` which sends

`G` to infinity. This lets us specify which extension of the 5-adic

valuation we care about in the above example::

sage: R.<x> = QQ[]

sage: v = K.valuation(GaussValuation(R, QQ.valuation(5)).augmentation(x + 2, infinity))

sage: w = K.valuation(GaussValuation(R, QQ.valuation(5)).augmentation(x + 1/2, infinity))

sage: v == w

False

Note that you get the same valuation, even if you write down the

pseudo-valuation differently::

sage: ww = K.valuation(GaussValuation(R, QQ.valuation(5)).augmentation(x + 3, infinity))

sage: w is ww

True

The valuation ``prime`` does not need to send the defining polynomial `G`

to infinity. It is sufficient if it singles out one of the valuations on

the number field. This is important if the prime only factors over the

completion, i.e., if it is not possible to write down one of the factors

within the number field::

sage: v = GaussValuation(R, QQ.valuation(5)).augmentation(x + 3, 1)

sage: K.valuation(v)

[ 5-adic valuation, v(x + 3) = 1 ]-adic valuation

Finally, ``prime`` can also be a fractional ideal of a number field if it

singles out an extension of a `p`-adic valuation of the base field::

sage: K.valuation(K.fractional_ideal(a + 1))

2-adic valuation

.. SEEALSO::

:meth:`Order.valuation() <sage.rings.number_field.order.Order.valuation>`,

:meth:`pAdicGeneric.valuation() <sage.rings.padics.padic_generic.pAdicGeneric.valuation>`

"""

from sage.rings.padics.padic_valuation import pAdicValuation

return pAdicValuation(self, prime)

def some_elements(self):

"""

Return a list of elements in the given number field.

EXAMPLES::

sage: R.<t> = QQ[]

sage: K.<a> = QQ.extension(t^2 - 2); K

Number Field in a with defining polynomial t^2 - 2

sage: K.some_elements()

[1, a, 2*a, 3*a - 4, 1/2, 1/3*a, 1/6*a, 0, 1/2*a, 2, ..., 12, -12*a + 18]

sage: T. = K[]

sage: M. = K.extension(t^3 - 5); M

Number Field in b with defining polynomial t^3 - 5 over its base field

sage: M.some_elements()

[1, b, 1/2*a*b, ..., 2/5*b^2 + 2/5, 1/6*b^2 + 5/6*b + 13/6, 2]

TESTS:

This also works in trivial extensions::

sage: R.<t> = QQ[]

sage: K.<a> = QQ.extension(t); K

Number Field in a with defining polynomial t

sage: K.some_elements()

[0, 1, 2, -1, 1/2, -1/2, 1/4, -2, 4]

"""

elements = []

polynomials = [self(f) for f in self.polynomial_ring().some_elements()]

for numerator in polynomials:

for denominator in polynomials:

if denominator:

some_element = numerator/denominator

if some_element not in elements:

elements.append(some_element)

return elements

class NumberField_absolute(NumberField_generic):

def __init__(self, polynomial, name, latex_name=None, check=True, embedding=None,

assume_disc_small=False, maximize_at_primes=None, structure=None):

"""

Function to initialize an absolute number field.

EXAMPLES::

sage: K = NumberField(x^17 + 3, 'a'); K

Number Field in a with defining polynomial x^17 + 3

sage: type(K)

sage: TestSuite(K).run()

"""

NumberField_generic.__init__(self, polynomial, name, latex_name, check, embedding,

assume_disc_small=assume_disc_small, maximize_at_primes=maximize_at_primes, structure=structure)

self._element_class = number_field_element.NumberFieldElement_absolute

self._zero_element = self._element_class(self, 0)

self._one_element = self._element_class(self, 1)

self._init_embedding_approx()

def _coerce_from_other_number_field(self, x):

"""

Coerce a number field element x into this number field.

Unless `x` is in ``QQ``, this requires ``x.parent()`` and

``self`` to have compatible embeddings: either they both embed

in a common field, or there is an embedding of ``x.parent()``

into ``self`` or the other way around. If no compatible

embeddings are found and `x` is not in ``QQ``, then raise

``TypeError``. This guarantees that these conversions respect

the field operations and conversions between several fields

commute.

REMARK:

The name of this method was chosen for historical reasons.

In fact, what it does is not a coercion but a conversion.

INPUT:

``x`` -- an element of some number field

OUTPUT:

An element of ``self`` corresponding to ``x``.

EXAMPLES::

sage: K.<a> = NumberField(x^3 + 2)

sage: L. = NumberField(x^2 + 1)

sage: K._coerce_from_other_number_field(L(2/3))

2/3

sage: L._coerce_from_other_number_field(K(0))

sage: K._coerce_from_other_number_field(b)

Traceback (most recent call last):

...

TypeError: No compatible natural embeddings found for Number Field in a with defining polynomial x^3 + 2 and Number Field in b with defining polynomial x^2 + 1

Two number fields both containing `i`::

sage: K.<a> = NumberField(x^4 + 6*x^2 + 1, embedding = CC(-2.4*I))

sage: L. = NumberField(x^4 + 8*x^2 + 4, embedding = CC(2.7*I))

sage: Ki = 1/2*a^3 + 5/2*a; Ki.minpoly()

x^2 + 1

sage: L(Ki)

-1/4*b^3 - 3/2*b

sage: K(L(Ki)) == Ki

True

sage: Q. = QuadraticField(-1)

sage: Q(Ki)

sage: Q(L(Ki))

sage: L( (Ki+2)^1000 )

737533628...075020804*b^3 + 442520177...450124824*b + 793311113...453515313

This fails if we don't specify the embeddings::

sage: K.<a> = NumberField(x^4 + 6*x^2 + 1)

sage: L. = NumberField(x^4 + 8*x^2 + 4)

sage: L(1/2*a^3 + 5/2*a)

Traceback (most recent call last):

...

TypeError: No compatible natural embeddings found for Number Field in b with defining polynomial x^4 + 8*x^2 + 4 and Number Field in a with defining polynomial x^4 + 6*x^2 + 1

Embeddings can also be `p`-adic::

sage: F = Qp(73)

sage: K.<a> = NumberField(x^4 + 6*x^2 + 1, embedding = F(1290990671961076190983179596556712119))

sage: L. = NumberField(x^4 + 8*x^2 + 4, embedding = F(1773398470280167815153042237103591466))

sage: L(2*a^3 + 10*a + 3)

b^3 + 6*b + 3

If we take the same non-Galois number field with two different

embeddings, conversion fails::

sage: K.<a> = NumberField(x^3 - 4*x + 1, embedding = 0.254)

sage: L. = NumberField(x^3 - 4*x + 1, embedding = 1.86)

sage: L(a)

Traceback (most recent call last):

...

ValueError: Cannot convert a to Number Field in b with defining polynomial x^3 - 4*x + 1 (using the specified embeddings)

Subfields automatically come with an embedding::

sage: K.<a> = NumberField(x^2 - 5)

sage: L., phi = K.subfield(-a)

sage: phi(b)

-a

sage: K(b)

-a

sage: L(a)

-b

Below we create two subfields of `K` which both contain `i`.

Since `L2` and `L3` both embed in `K`, conversion works::

sage: K.<z> = NumberField(x^8 - x^4 + 1)

sage: i = (x^2+1).roots(ring=K)[0][0]

sage: r2 = (x^2-2).roots(ring=K)[0][0]

sage: r3 = (x^2-3).roots(ring=K)[0][0]

sage: L2.<a2>, phi2 = K.subfield(r2+i)

sage: L3.<a3>, phi3 = K.subfield(r3+i)

sage: i_in_L2 = L2(i); i_in_L2

1/6*a2^3 + 1/6*a2

sage: i_in_L3 = L3(i); i_in_L3

1/8*a3^3

sage: L2(i_in_L3) == i_in_L2

True

sage: L3(i_in_L2) == i_in_L3

True

TESTS:

The following was fixed in :trac:`8800`::

sage: P.<x> = QQ[]

sage: K.<a> = NumberField(x^3-5,embedding=0)

sage: L. = K.extension(x^2+a)

sage: F,R = L.construction()

sage: F(R) == L #indirect doctest

True

AUTHORS:

- Jeroen Demeyer (2011-09-30): :trac:`11869`

"""

# Special case for x in QQ. This is common, so should be fast.

xpol = x.polynomial()

if xpol.degree() <= 0:

return self._element_class(self, xpol[0])

# Convert from L to K

K = self

L = x.parent()

# Find embeddings for K and L. If no embedding is given, simply

# take the identity map as "embedding". This handles the case

# where one field is created as subfield of the other.

Kgen = K.gen_embedding()

if Kgen is None:

Kgen = K.gen()

KF = Kgen.parent()

Lgen = L.gen_embedding()

if Lgen is None:

Lgen = L.gen()

LF = Lgen.parent()

# Do not use CDF or RDF because of constraints on the

# exponent of floating-point numbers

from sage.rings.all import RealField, ComplexField

CC = ComplexField(53)

RR = RealField(53)

# Find a common field F into which KF and LF both embed.

if CC.has_coerce_map_from(KF) and CC.has_coerce_map_from(LF):

# We postpone converting Kgen and Lgen to F until we know the

# floating-point precision required.

F = CC

elif KF is LF:

F = KF

elif KF.has_coerce_map_from(LF):

F = KF

Lgen = F(Lgen)

elif LF.has_coerce_map_from(KF):

F = LF

Kgen = F(Kgen)

else:

raise TypeError("No compatible natural embeddings found for %s and %s"%(KF,LF))

# List of candidates for K(x)

f = x.minpoly()

ys = f.roots(ring=K, multiplicities=False)

if not ys:

raise ValueError("Cannot convert %s to %s (regardless of embeddings)"%(x,K))

# Define a function are_roots_equal to determine whether two

# roots of f are equal. A simple a == b does not suffice for

# inexact fields because of floating-point errors.

if F.is_exact():

are_roots_equal = lambda a,b: a == b

else:

### Compute a lower bound on the distance between the roots of f.

### This essentially gives the precision to work with.

# A function

# log2abs: F --> RR

# x |-> log2(abs(x))

# This should work for all fields F with an absolute value.

# The p-adic absolute value goes into QQ, so we need the RR().

log2abs = lambda x: RR(F(x).abs()).log2()

# Compute half Fujiwara's bound on the roots of f

n = f.degree()

log_half_root_bound = log2abs(f[0]/2)/n

for i in range(1,n):

bd = log2abs(f[i])/(n-i)

if bd > log_half_root_bound:

log_half_root_bound = bd

# Twice the bound on the roots of f, in other words an upper

# bound for the distance between two roots.

log_double_root_bound = log_half_root_bound + 2.0 # 2.0 = log2(4)

# Now we compute the minimum distance between two roots of f

# using the fact that the discriminant of f is the product of

# all root distances.

# We use pari to compute the discriminant to work around #11872.

log_root_diff = log2abs(pari(f).poldisc())*0.5 - (n*(n-1)*0.5 - 1.0)*log_double_root_bound

# Let eps be 1/128 times the minimal root distance.

# This implies: If two roots of f are at distance <= eps, then

# they are equal. The factor 128 is arbitrary, it is an extra

# safety margin.

eps = (log_root_diff - 7.0).exp2()

are_roots_equal = lambda a,b: (a-b).abs() <= eps

if F is CC:

# Adjust the precision of F, sufficient to represent all

# the temporaries in the computation with a precision

# of eps, plus some extra bits.

H = [log_double_root_bound - 1.0]

for e in [x] + ys:

H += [log2abs(c) for c in e.polynomial().coefficients()]

prec = (max(H) + RR(n+1).log2() - log_root_diff).ceil() + 12 + n

F = ComplexField(prec=prec)

Kgen = F(Kgen)

Lgen = F(Lgen)

# Embed x and the y's in F

emb_x = x.polynomial()(Lgen)

for y in ys:

emb_y = y.polynomial()(Kgen)

if are_roots_equal(emb_x, emb_y):

return y

raise ValueError("Cannot convert %s to %s (using the specified embeddings)"%(x,K))

def _coerce_map_from_(self, R):

"""

Canonical coercion of a ring R into self.

Currently any ring coercing into the base ring canonically coerces

into this field, as well as orders in any number field coercing into

this field, and of course the field itself as well.

Two embedded number fields may mutually coerce into each other, if

the pushout of the two ambient fields exists and if it is possible

to construct an :class:`~sage.rings.number_field.number_field_morphisms.EmbeddedNumberFieldMorphism`.

EXAMPLES::

sage: S.<y> = NumberField(x^3 + x + 1)

sage: S.coerce(int(4)) # indirect doctest

sage: S.coerce(long(7))

sage: S.coerce(-Integer(2))

-2

sage: z = S.coerce(-7/8); z, type(z)

(-7/8, <type 'sage.rings.number_field.number_field_element.NumberFieldElement_absolute'>)

sage: S.coerce(y) is y

True

Fields with embeddings into an ambient field coerce naturally by the given embedding::

sage: CyclotomicField(15).coerce(CyclotomicField(5).0 - 17/3)

zeta15^3 - 17/3

sage: K.<a> = CyclotomicField(16)

sage: K(CyclotomicField(4).0)

a^4

sage: QuadraticField(-3, 'a').coerce_map_from(CyclotomicField(3))

Generic morphism:

From: Cyclotomic Field of order 3 and degree 2

To: Number Field in a with defining polynomial x^2 + 3

Defn: zeta3 -> 1/2*a - 1/2

Two embedded number fields with mutual coercions (testing against a

bug that was fixed in :trac:`8800`)::

sage: K.<r4> = NumberField(x^4-2)

sage: L1.<r2_1> = NumberField(x^2-2, embedding = r4**2)

sage: L2.<r2_2> = NumberField(x^2-2, embedding = -r4**2)

sage: r2_1+r2_2 # indirect doctest

sage: (r2_1+r2_2).parent() is L1

True

sage: (r2_2+r2_1).parent() is L2

True

Coercion of an order (testing against a bug that was fixed in

:trac:`8800`)::

sage: K.has_coerce_map_from(L1)

True

sage: L1.has_coerce_map_from(K)

False

sage: K.has_coerce_map_from(L1.maximal_order())

True

sage: L1.has_coerce_map_from(K.maximal_order())

False

There are situations for which one might imagine conversion

could make sense (at least after fixing choices), but of course

there will be no coercion from the Symbolic Ring to a Number Field::

sage: K.<a> = QuadraticField(2)

sage: K.coerce(sqrt(2))

Traceback (most recent call last):

...

TypeError: no canonical coercion from Symbolic Ring to Number Field in a with defining polynomial x^2 - 2

TESTS::

sage: K.<a> = NumberField(polygen(QQ)^3-2)

sage: type(K.coerce_map_from(QQ))

Make sure we still get our optimized morphisms for special fields::

sage: K.<a> = NumberField(polygen(QQ)^2-2)

sage: type(K.coerce_map_from(QQ))

"""

if R in integer_types:

return self._generic_coerce_map(R)

elif R in (ZZ, QQ, self.base()):

return self._generic_coerce_map(R)

from sage.rings.number_field.order import is_NumberFieldOrder

if is_NumberFieldOrder(R) and self.has_coerce_map_from(R.number_field()):

return self._generic_coerce_map(R)

# R is not QQ by the above tests

if is_NumberField(R) and R.coerce_embedding() is not None:

if self.coerce_embedding() is not None:

try:

return number_field_morphisms.EmbeddedNumberFieldMorphism(R, self)

except ValueError: # no common embedding found

return None

else:

# R is embedded, self isn't. So, we could only have

# the forgetful coercion. But this yields to non-commuting

# coercions, as was pointed out at ticket #8800

return None

def base_field(self):

"""

Returns the base field of self, which is always QQ

EXAMPLES::

sage: K = CyclotomicField(5)

sage: K.base_field()

Rational Field

"""

return QQ

def is_absolute(self):

"""

Returns True since self is an absolute field.

EXAMPLES::

sage: K = CyclotomicField(5)

sage: K.is_absolute()

True

"""

return True

def absolute_polynomial(self):

r"""

Return absolute polynomial that defines this absolute field. This

is the same as ``self.polynomial()``.

EXAMPLES::

sage: K.<a> = NumberField(x^2 + 1)

sage: K.absolute_polynomial ()

x^2 + 1

"""

return self.polynomial()

def absolute_generator(self):

r"""

An alias for

:meth:`sage.rings.number_field.number_field.NumberField_generic.gen`.

This is provided for consistency with relative fields, where the

element returned by

:meth:`sage.rings.number_field.number_field_rel.NumberField_relative.gen`

only generates the field over its base field (not necessarily over

`\QQ`).

EXAMPLES::

sage: K.<a> = NumberField(x^2 - 17)

sage: K.absolute_generator()

"""

return self.gen()

def optimized_representation(self, name=None, both_maps=True):

"""

Return a field isomorphic to self with a better defining polynomial

if possible, along with field isomorphisms from the new field to

self and from self to the new field.

EXAMPLES: We construct a compositum of 3 quadratic fields, then

find an optimized representation and transform elements back and

forth.

sage: K = NumberField([x^2 + p for p in [5, 3, 2]],'a').absolute_field('b'); K

Number Field in b with defining polynomial x^8 + 40*x^6 + 352*x^4 + 960*x^2 + 576

sage: L, from_L, to_L = K.optimized_representation()

sage: L # your answer may different, since algorithm is random

Number Field in b1 with defining polynomial x^8 + 4*x^6 + 7*x^4 +

36*x^2 + 81

sage: to_L(K.0) # random

4/189*b1^7 + 1/63*b1^6 + 1/27*b1^5 - 2/9*b1^4 - 5/27*b1^3 - 8/9*b1^2 + 3/7*b1 - 3/7

sage: from_L(L.0) # random

1/1152*b^7 - 1/192*b^6 + 23/576*b^5 - 17/96*b^4 + 37/72*b^3 - 5/6*b^2 + 55/24*b - 3/4

The transformation maps are mutually inverse isomorphisms.

sage: from_L(to_L(K.0)) == K.0

True

sage: to_L(from_L(L.0)) == L.0

True

Number fields defined by non-monic and non-integral

polynomials are supported (:trac:`252`)::

sage: K.<a> = NumberField(7/9*x^3 + 7/3*x^2 - 56*x + 123)

sage: K.optimized_representation()

(Number Field in a1 with defining polynomial x^3 - 7*x - 7,

Ring morphism:

From: Number Field in a1 with defining polynomial x^3 - 7*x - 7

To: Number Field in a with defining polynomial 7/9*x^3 + 7/3*x^2 - 56*x + 123

Defn: a1 |--> 7/225*a^2 - 7/75*a - 42/25,

Ring morphism:

From: Number Field in a with defining polynomial 7/9*x^3 + 7/3*x^2 - 56*x + 123

To: Number Field in a1 with defining polynomial x^3 - 7*x - 7

Defn: a |--> -15/7*a1^2 + 9)

"""

if name is None:

name = self.variable_names()

name = normalize_names(1, name)[0]

f = self.absolute_polynomial().__pari__()

g, alpha = f.polredbest(flag=1)

beta = alpha.modreverse()

b = self(QQ['x'](lift(beta)))

h = QQ['x'](g)

embedding = None

if self.coerce_embedding() is not None:

embedding = self.coerce_embedding()(b)

# trac 7695 add a _ to prevent zeta70 etc.

if name[-1].isdigit():

new_name = name + '_1'

else:

new_name = name + '1'

K = NumberField(h, names=new_name, embedding=embedding)

from_K = K.hom([b])

if both_maps:

a = K(alpha)

to_K = self.hom([a])

return K, from_K, to_K

return K, from_K

def optimized_subfields(self, degree=0, name=None, both_maps=True):

"""

Return optimized representations of many (but *not* necessarily

all!) subfields of self of the given degree, or of all possible degrees if

degree is 0.

EXAMPLES::

sage: K = NumberField([x^2 + p for p in [5, 3, 2]],'a').absolute_field('b'); K

Number Field in b with defining polynomial x^8 + 40*x^6 + 352*x^4 + 960*x^2 + 576

sage: L = K.optimized_subfields(name='b')

sage: L[0][0]

Number Field in b0 with defining polynomial x

sage: L[1][0]

Number Field in b1 with defining polynomial x^2 - 3*x + 3

sage: [z[0] for z in L] # random -- since algorithm is random

[Number Field in b0 with defining polynomial x - 1,

Number Field in b1 with defining polynomial x^2 - x + 1,

Number Field in b2 with defining polynomial x^4 - 5*x^2 + 25,

Number Field in b3 with defining polynomial x^4 - 2*x^2 + 4,

Number Field in b4 with defining polynomial x^8 + 4*x^6 + 7*x^4 + 36*x^2 + 81]

We examine one of the optimized subfields in more detail::

sage: M, from_M, to_M = L[2]

sage: M # random

Number Field in b2 with defining polynomial x^4 - 5*x^2 + 25

sage: from_M # may be slightly random

Ring morphism:

From: Number Field in b2 with defining polynomial x^4 - 5*x^2 + 25

To: Number Field in a1 with defining polynomial x^8 + 40*x^6 + 352*x^4 + 960*x^2 + 576

Defn: b2 |--> -5/1152*a1^7 + 1/96*a1^6 - 97/576*a1^5 + 17/48*a1^4 - 95/72*a1^3 + 17/12*a1^2 - 53/24*a1 - 1

The to_M map is None, since there is no map from K to M::

sage: to_M

We apply the from_M map to the generator of M, which gives a

rather large element of `K`::

sage: from_M(M.0) # random

-5/1152*a1^7 + 1/96*a1^6 - 97/576*a1^5 + 17/48*a1^4 - 95/72*a1^3 + 17/12*a1^2 - 53/24*a1 - 1

Nevertheless, that large-ish element lies in a degree 4 subfield::

sage: from_M(M.0).minpoly() # random

x^4 - 5*x^2 + 25

TESTS:

Number fields defined by non-monic and non-integral

polynomials are supported (:trac:`252`)::

sage: K.<a> = NumberField(2*x^4 + 6*x^2 + 1/2)

sage: K.optimized_subfields()

[

(Number Field in a0 with defining polynomial x, Ring morphism:

From: Number Field in a0 with defining polynomial x

To: Number Field in a with defining polynomial 2*x^4 + 6*x^2 + 1/2

Defn: 0 |--> 0, None),

(Number Field in a1 with defining polynomial x^2 - 2*x + 2, Ring morphism:

From: Number Field in a1 with defining polynomial x^2 - 2*x + 2

To: Number Field in a with defining polynomial 2*x^4 + 6*x^2 + 1/2

Defn: a1 |--> a^3 + 7/2*a + 1, None),

(Number Field in a2 with defining polynomial x^2 - 2*x + 2, Ring morphism:

From: Number Field in a2 with defining polynomial x^2 - 2*x + 2

To: Number Field in a with defining polynomial 2*x^4 + 6*x^2 + 1/2

Defn: a2 |--> -a^3 - 7/2*a + 1, None),

(Number Field in a3 with defining polynomial x^2 - 2, Ring morphism:

From: Number Field in a3 with defining polynomial x^2 - 2

To: Number Field in a with defining polynomial 2*x^4 + 6*x^2 + 1/2

Defn: a3 |--> a^2 + 3/2, None),

(Number Field in a4 with defining polynomial x^2 + 1, Ring morphism:

From: Number Field in a4 with defining polynomial x^2 + 1

To: Number Field in a with defining polynomial 2*x^4 + 6*x^2 + 1/2

Defn: a4 |--> a^3 + 7/2*a, None),

(Number Field in a5 with defining polynomial x^2 + 2, Ring morphism:

From: Number Field in a5 with defining polynomial x^2 + 2

To: Number Field in a with defining polynomial 2*x^4 + 6*x^2 + 1/2

Defn: a5 |--> 2*a^3 + 5*a, None),

(Number Field in a6 with defining polynomial x^4 + 1, Ring morphism:

From: Number Field in a6 with defining polynomial x^4 + 1

To: Number Field in a with defining polynomial 2*x^4 + 6*x^2 + 1/2

Defn: a6 |--> a^3 + 1/2*a^2 + 5/2*a + 3/4, Ring morphism:

From: Number Field in a with defining polynomial 2*x^4 + 6*x^2 + 1/2

To: Number Field in a6 with defining polynomial x^4 + 1

Defn: a |--> -1/2*a6^3 + a6^2 - 1/2*a6)

]

"""

return self._subfields_helper(degree=degree,name=name,

both_maps=both_maps,optimize=True)

def change_names(self, names):

r"""

Return number field isomorphic to self but with the given generator

name.

INPUT:

- ``names`` - should be exactly one variable name.

Also, ``K.structure()`` returns from_K and to_K,

where from_K is an isomorphism from K to self and to_K is an

isomorphism from self to K.

EXAMPLES::

sage: K.<z> = NumberField(x^2 + 3); K

Number Field in z with defining polynomial x^2 + 3

sage: L.<ww> = K.change_names()

sage: L

Number Field in ww with defining polynomial x^2 + 3

sage: L.structure()[0]

Isomorphism given by variable name change map:

From: Number Field in ww with defining polynomial x^2 + 3

To: Number Field in z with defining polynomial x^2 + 3

sage: L.structure()[0](ww + 5/3)

z + 5/3

"""

return self.absolute_field(names)

def subfields(self, degree=0, name=None):

"""

Return all subfields of self of the given degree,

or of all possible degrees if degree is 0. The subfields are returned as

absolute fields together with an embedding into self. For the case of the

field itself, the reverse isomorphism is also provided.

EXAMPLES::

sage: K.<a> = NumberField( [x^3 - 2, x^2 + x + 1] )

sage: K = K.absolute_field('b')

sage: S = K.subfields()

sage: len(S)

sage: [k[0].polynomial() for k in S]

[x - 3,

x^2 - 3*x + 9,

x^3 - 3*x^2 + 3*x + 1,

x^3 - 3*x^2 + 3*x - 17,

x^6 - 3*x^5 + 6*x^4 - 11*x^3 + 12*x^2 + 3*x + 1]

sage: R.<t> = QQ[]

sage: L = NumberField(t^3 - 3*t + 1, 'c')

sage: [k[1] for k in L.subfields()]

[Ring morphism:

From: Number Field in c0 with defining polynomial t

To: Number Field in c with defining polynomial t^3 - 3*t + 1

Defn: 0 |--> 0,

Ring morphism:

From: Number Field in c1 with defining polynomial t^3 - 3*t + 1

To: Number Field in c with defining polynomial t^3 - 3*t + 1

Defn: c1 |--> c]

sage: len(L.subfields(2))

sage: len(L.subfields(1))

TESTS:

Number fields defined by non-monic and non-integral

polynomials are supported (:trac:`252`)::

sage: K.<a> = NumberField(2*x^4 + 6*x^2 + 1/2)

sage: K.subfields()

[

(Number Field in a0 with defining polynomial x, Ring morphism:

From: Number Field in a0 with defining polynomial x

To: Number Field in a with defining polynomial 2*x^4 + 6*x^2 + 1/2

Defn: 0 |--> 0, None),

(Number Field in a1 with defining polynomial x^2 - 2, Ring morphism:

From: Number Field in a1 with defining polynomial x^2 - 2

To: Number Field in a with defining polynomial 2*x^4 + 6*x^2 + 1/2

Defn: a1 |--> a^2 + 3/2, None),

(Number Field in a2 with defining polynomial x^2 + 4, Ring morphism:

From: Number Field in a2 with defining polynomial x^2 + 4

To: Number Field in a with defining polynomial 2*x^4 + 6*x^2 + 1/2

Defn: a2 |--> 2*a^3 + 7*a, None),

(Number Field in a3 with defining polynomial x^2 + 2, Ring morphism:

From: Number Field in a3 with defining polynomial x^2 + 2

To: Number Field in a with defining polynomial 2*x^4 + 6*x^2 + 1/2

Defn: a3 |--> 2*a^3 + 5*a, None),

(Number Field in a4 with defining polynomial x^4 + 1, Ring morphism:

From: Number Field in a4 with defining polynomial x^4 + 1

To: Number Field in a with defining polynomial 2*x^4 + 6*x^2 + 1/2

Defn: a4 |--> a^3 + 1/2*a^2 + 5/2*a + 3/4, Ring morphism:

From: Number Field in a with defining polynomial 2*x^4 + 6*x^2 + 1/2

To: Number Field in a4 with defining polynomial x^4 + 1

Defn: a |--> -1/2*a4^3 + a4^2 - 1/2*a4)

]

"""

return self._subfields_helper(degree=degree, name=name,

both_maps=True, optimize=False)

def _subfields_helper(self, degree=0, name=None, both_maps=True, optimize=False):

"""

Internal function: common code for optimized_subfields() and subfields().

TESTS:

Let's make sure embeddings are being respected::

sage: K.<a> = NumberField(x^4 - 23, embedding=50)

sage: K, CDF(a)

(Number Field in a with defining polynomial x^4 - 23, 2.1899387030948425)

sage: Ss = K.subfields(); len(Ss) # indirect doctest

sage: diffs = [ S.coerce_embedding()(S.gen()) - CDF(S_into_K(S.gen())) for S, S_into_K, _ in Ss ]

sage: all(abs(diff) < 1e-5 for diff in diffs)

True

sage: L1, _, _ = K.subfields(2)[0]; L1, CDF(L1.gen()) # indirect doctest

(Number Field in a0 with defining polynomial x^2 - 23, -4.795831523312719)

If we take a different embedding of the large field, we get a

different embedding of the degree 2 subfield::

sage: K.<a> = NumberField(x^4 - 23, embedding=-50)

sage: L2, _, _ = K.subfields(2)[0]; L2, CDF(L2.gen()) # indirect doctest

(Number Field in a0 with defining polynomial x^2 - 23, -4.795831523312719)

Test for :trac:`7695`::

sage: F = CyclotomicField(7)

sage: K = F.subfields(3)[0][0]

sage: K

Number Field in zeta7_0 with defining polynomial x^3 + x^2 - 2*x - 1

"""

if name is None:

name = self.variable_names()

name = normalize_names(1, name)[0]

try:

return self.__subfields[name, degree, both_maps, optimize]

except AttributeError:

self.__subfields = {}

except KeyError:

pass

f = self.pari_polynomial()

if optimize:

v = f.polred(2)

elts = v[0]

polys = v[1]

else:

v = f.nfsubfields(degree)

elts = [x[1] for x in v]

polys = [x[0] for x in v]

R = self.polynomial_ring()

embedding = None

ans = []

for i in range(len(elts)):

f = R(polys[i])

if not (degree == 0 or f.degree() == degree):

continue

a = self(elts[i], check=False)

if self.coerce_embedding() is not None:

embedding = self.coerce_embedding()(a)

# trac 7695 add a _ to prevent zeta70 etc.

if name[-1].isdigit():

new_name= name+ '_' + str(i)

else:

new_name = name + str(i)

K = NumberField(f, names=new_name, embedding=embedding)

from_K = K.hom([a]) # check=False here ?? would be safe unless there are bugs.

if both_maps and K.degree() == self.degree():

g = K['x'](self.polynomial())

a = from_K(K.gen())

for root in g.roots(multiplicities=False):

to_K = self.hom([root]) # check=False here ??

if to_K(a) == K.gen():

break

else:

to_K = None

ans.append((K, from_K, to_K))

ans = Sequence(ans, immutable=True, cr=ans!=[])

self.__subfields[name, degree, both_maps, optimize] = ans

return ans

def maximal_order(self, v=None):

"""

Return the maximal order, i.e., the ring of integers, associated to

this number field.

INPUT:

- ``v`` - (default: ``None``) ``None``, a prime, or a list of primes.

- if ``v`` is ``None``, return the maximal order.

- if ``v`` is a prime, return an order that is `p`-maximal.

- if ``v`` is a list, return an order that is maximal at each prime

in the list ``v``.

EXAMPLES:

In this example, the maximal order cannot be generated by a single

element::

sage: k.<a> = NumberField(x^3 + x^2 - 2*x+8)

sage: o = k.maximal_order()

sage: o

Maximal Order in Number Field in a with defining polynomial x^3 + x^2 - 2*x + 8

We compute `p`-maximal orders for several `p`. Note

that computing a `p`-maximal order is much faster in

general than computing the maximal order::

sage: p = next_prime(10^22); q = next_prime(10^23)

sage: K.<a> = NumberField(x^3 - p*q)

sage: K.maximal_order([3]).basis()

[1/3*a^2 + 1/3*a + 1/3, a, a^2]

sage: K.maximal_order([2]).basis()

[1, a, a^2]

sage: K.maximal_order([p]).basis()

[1, a, a^2]

sage: K.maximal_order([q]).basis()

[1, a, a^2]

sage: K.maximal_order([p,3]).basis()

[1/3*a^2 + 1/3*a + 1/3, a, a^2]

An example with bigger discriminant::

sage: p = next_prime(10^97); q = next_prime(10^99)

sage: K.<a> = NumberField(x^3 - p*q)

sage: K.maximal_order(prime_range(10000)).basis()

[1, a, a^2]

"""

return self._maximal_order(self._normalize_prime_list(v))

@cached_method

def _maximal_order(self, v):

r"""

Helper method which adds caching to :meth:`maximal_order`.

EXAMPLES::

sage: k.<a> = NumberField(x^3 + x^2 - 2*x+8)

sage: k.maximal_order() is k.maximal_order() # indirect doctest

True

TESTS:

Number fields defined by non-monic and non-integral

polynomials are supported (:trac:`252`)::

sage: K.<a> = NumberField(3*x^2 + 1)

sage: K.maximal_order().basis()

[3/2*a + 1/2, 3*a]

"""

B = [self(b, check=False) for b in self._pari_integral_basis(v=v)]

import sage.rings.number_field.order as order

return order.absolute_order_from_module_generators(B,

check_integral=False, check_rank=False,

check_is_ring=False, is_maximal=not v)

def order(self, *args, **kwds):

r"""

Return the order with given ring generators in the maximal order of

this number field.

INPUT:

- ``gens`` - list of elements in this number field; if no generators

are given, just returns the cardinality of this number field

(`\infty`) for consistency.

- ``check_is_integral`` - bool (default: ``True``), whether to check

that each generator is integral.

- ``check_rank`` - bool (default: ``True``), whether to check that the

ring generated by ``gens`` is of full rank.

- ``allow_subfield`` - bool (default: ``False``), if ``True`` and the

generators do not generate an order, i.e., they generate a subring

of smaller rank, instead of raising an error, return an order in a

smaller number field.

EXAMPLES::

sage: k. = NumberField(x^2 + 1)

sage: k.order(2*i)

Order in Number Field in i with defining polynomial x^2 + 1

sage: k.order(10*i)

Order in Number Field in i with defining polynomial x^2 + 1

sage: k.order(3)

Traceback (most recent call last):

...

ValueError: the rank of the span of gens is wrong

sage: k.order(i/2)

Traceback (most recent call last):

...

ValueError: each generator must be integral

Alternatively, an order can be constructed by adjoining elements to

`\ZZ`::

sage: K.<a> = NumberField(x^3 - 2)

sage: ZZ[a]

Order in Number Field in a0 with defining polynomial x^3 - 2

TESTS:

We verify that :trac:`2480` is fixed::

sage: K.<a> = NumberField(x^4 + 4*x^2 + 2)

sage: B = K.integral_basis()

sage: K.order(*B)

Order in Number Field in a with defining polynomial x^4 + 4*x^2 + 2

sage: K.order(B)

Order in Number Field in a with defining polynomial x^4 + 4*x^2 + 2

sage: K.order(gens=B)

Order in Number Field in a with defining polynomial x^4 + 4*x^2 + 2

"""

# set gens appropriately from the arguments

gens = kwds.pop('gens', args)

if len(gens) == 0:

return NumberField_generic.order(self)

if len(gens) == 1 and isinstance(gens[0], (list, tuple)):

gens = gens[0]

gens = map(self, gens)

return self._order(tuple(gens), **kwds)

@cached_method

def _order(self, gens, **kwds):

r"""

Helper method for :meth:`order` which adds caching. See :meth:`order`

for a description of the parameters and keyword parameters.

TESTS:

Test that caching works::

sage: K.<a> = NumberField(x^3 - 2)

sage: K.order(a) is K.order(a) # indirect doctest

True

Keywords have no influence on the caching::

sage: K.order(a) is K.order(a,check_is_integral=True) is K.order(a,check_is_integral=False)

True

Even if the order lives in a different field, caching works (currently,

however, ``allow_subfield`` is incorrect :trac:`16046`)::

sage: K.<a> = NumberField(x**4+3)

sage: o = K.order([a**2], allow_subfield=True)

sage: o is K.order([a**2], allow_subfield=True)

True

Different generators for the same order::

sage: K.order(a) is K.order(a,a^2) is K.order(a^2,a)

True

"""

import sage.rings.number_field.order as order

ret = order.absolute_order_from_ring_generators(gens, **kwds)

# we make sure that the result is a unique parent even if it the order

# lives in a different field

if ret.ambient() is not self:

return ret.ambient().order(gens, **kwds)

gens = ret.basis()

if self._order.is_in_cache(gens):

# different ways of specifying the same set of generators lead to

# the same order - this is to make sure that orders are unique

# parents

return self._order(gens)

self._order.set_cache(ret, gens)

return ret

def vector_space(self):

"""

Return a vector space V and isomorphisms self --> V and V --> self.

OUTPUT:

- ``V`` - a vector space over the rational numbers

- ``from_V`` - an isomorphism from V to self

- ``to_V`` - an isomorphism from self to V

EXAMPLES::

sage: k.<a> = NumberField(x^3 + 2)

sage: V, from_V, to_V = k.vector_space()

sage: from_V(V([1,2,3]))

3*a^2 + 2*a + 1

sage: to_V(1 + 2*a + 3*a^2)

(1, 2, 3)

sage: V

Vector space of dimension 3 over Rational Field

sage: to_V

Isomorphism map:

From: Number Field in a with defining polynomial x^3 + 2

To: Vector space of dimension 3 over Rational Field

sage: from_V(to_V(2/3*a - 5/8))

2/3*a - 5/8

sage: to_V(from_V(V([0,-1/7,0])))

(0, -1/7, 0)

"""

try:

return self.__vector_space

except AttributeError:

V = QQ**self.degree()

from_V = maps.MapVectorSpaceToNumberField(V, self)

to_V = maps.MapNumberFieldToVectorSpace(self, V)

self.__vector_space = (V, from_V, to_V)

return self.__vector_space

def absolute_vector_space(self):

r"""

Return vector space over `\QQ` corresponding to this

number field, along with maps from that space to this number field

and in the other direction.

For an absolute extension this is identical to

``self.vector_space()``.

EXAMPLES::

sage: K.<a> = NumberField(x^3 - 5)

sage: K.absolute_vector_space()

(Vector space of dimension 3 over Rational Field,

Isomorphism map:

From: Vector space of dimension 3 over Rational Field

To: Number Field in a with defining polynomial x^3 - 5,

Isomorphism map:

From: Number Field in a with defining polynomial x^3 - 5

To: Vector space of dimension 3 over Rational Field)

"""

return self.vector_space()

def _galois_closure_and_embedding(self, names=None):

r"""

Return number field `K` that is the Galois closure of self and an

embedding of self into `K`.

INPUT:

- ``names`` - variable name for Galois closure

.. warning::

This is an internal function; see :meth:`galois_closure`.

EXAMPLES:

For medium-sized Galois groups of fields with small discriminants,

this computation is feasible::

sage: K.<a> = NumberField(x^6 + 4*x^2 + 2)

sage: K.galois_group(type='pari').order()

sage: L, phi = K._galois_closure_and_embedding('c')

sage: phi.domain() is K, phi.codomain() is L

(True, True)

sage: L

Number Field in c with defining polynomial x^48 + 8*x^46 - 20*x^44 - 520*x^42 + 12106*x^40 - 68344*x^38 + 463156*x^36 - 1823272*x^34 + 8984591*x^32 - 25016080*x^30 + 84949344*x^28 - 163504384*x^26 + 417511068*x^24 - 394687376*x^22 + 836352224*x^20 + 72845696*x^18 + 1884703919*x^16 + 732720520*x^14 + 3960878676*x^12 + 2507357768*x^10 + 5438373834*x^8 + 3888508744*x^6 + 4581432268*x^4 + 1765511400*x^2 + 1723993441

sage: K.defining_polynomial()( phi(K.gen()) )

"""

if names is None:

raise TypeError("You must specify the name of the generator.")

try:

# compose with variable renaming

L = self.__galois_closure.change_names(names)

L_to_orig, orig_to_L = L.structure()

# "flatten" the composition by hand

self_into_L = self.hom([ (orig_to_L * self.__galois_closure_embedding)(self.gen()) ])

return (L, self_into_L)

except AttributeError:

pass

# Compute degree of Galois closure if possible

try:

deg = self.galois_group(type='pari').order()

except NotImplementedError:

deg = None

L, self_into_L = self.defining_polynomial().change_ring(self).splitting_field(names, map=True, degree_multiple=deg)

self.__galois_closure = L

self.__galois_closure_embedding = self_into_L

return (self.__galois_closure, self.__galois_closure_embedding)

def galois_closure(self, names=None, map=False):

"""

Return number field `K` that is the Galois closure of self,

i.e., is generated by all roots of the defining polynomial of

self, and possibly an embedding of self into `K`.

INPUT:

- ``names`` - variable name for Galois closure

- ``map`` - (default: False) also return an embedding of self into `K`

EXAMPLES::

sage: K.<a> = NumberField(x^4 - 2)

sage: M = K.galois_closure('b'); M

Number Field in b with defining polynomial x^8 + 28*x^4 + 2500

sage: L.<a2> = K.galois_closure(); L

Number Field in a2 with defining polynomial x^8 + 28*x^4 + 2500

sage: K.galois_group(names=("a3")).order()

sage: phi = K.embeddings(L)[0]

sage: phi(K.0)

1/120*a2^5 + 19/60*a2

sage: phi(K.0).minpoly()

x^4 - 2

sage: L, phi = K.galois_closure('b', map=True)

sage: L

Number Field in b with defining polynomial x^8 + 28*x^4 + 2500

sage: phi

Ring morphism:

From: Number Field in a with defining polynomial x^4 - 2

To: Number Field in b with defining polynomial x^8 + 28*x^4 + 2500

Defn: a |--> 1/240*b^5 - 41/120*b

A cyclotomic field is already Galois::

sage: K.<a> = NumberField(cyclotomic_polynomial(23))

sage: L.<z> = K.galois_closure()

sage: L

Number Field in z with defining polynomial x^22 + x^21 + x^20 + x^19 + x^18 + x^17 + x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1

TESTS:

Let's make sure we're renaming correctly::

sage: K.<a> = NumberField(x^4 - 2)

sage: L, phi = K.galois_closure('cc', map=True)

sage: L

Number Field in cc with defining polynomial x^8 + 28*x^4 + 2500

sage: phi

Ring morphism:

From: Number Field in a with defining polynomial x^4 - 2

To: Number Field in cc with defining polynomial x^8 + 28*x^4 + 2500

Defn: a |--> 1/240*cc^5 - 41/120*cc

"""

L, self_into_L = self._galois_closure_and_embedding(names)

if map:

return (L, self_into_L)

else:

return L

def automorphisms(self):

r"""

Compute all Galois automorphisms of self.

This uses PARI's :pari:`nfgaloisconj` and is much faster than root finding

for many fields.

EXAMPLES::

sage: K.<a> = NumberField(x^2 + 10000)

sage: K.automorphisms()

[

Ring endomorphism of Number Field in a with defining polynomial x^2 + 10000

Defn: a |--> a,

Ring endomorphism of Number Field in a with defining polynomial x^2 + 10000

Defn: a |--> -a

]

Here's a larger example, that would take some time if we found

roots instead of using PARI's specialized machinery::

sage: K = NumberField(x^6 - x^4 - 2*x^2 + 1, 'a')

sage: len(K.automorphisms())

`L` is the Galois closure of `K`::

sage: L = NumberField(x^24 - 84*x^22 + 2814*x^20 - 15880*x^18 - 409563*x^16 - 8543892*x^14 + 25518202*x^12 + 32831026956*x^10 - 672691027218*x^8 - 4985379093428*x^6 + 320854419319140*x^4 + 817662865724712*x^2 + 513191437605441, 'a')

sage: len(L.automorphisms())

Number fields defined by non-monic and non-integral

polynomials are supported (:trac:`252`)::

sage: R.<x> = QQ[]

sage: f = 7/9*x^3 + 7/3*x^2 - 56*x + 123

sage: K.<a> = NumberField(f)

sage: A = K.automorphisms(); A

[

Ring endomorphism of Number Field in a with defining polynomial 7/9*x^3 + 7/3*x^2 - 56*x + 123

Defn: a |--> a,

Ring endomorphism of Number Field in a with defining polynomial 7/9*x^3 + 7/3*x^2 - 56*x + 123

Defn: a |--> -7/15*a^2 - 18/5*a + 96/5,

Ring endomorphism of Number Field in a with defining polynomial 7/9*x^3 + 7/3*x^2 - 56*x + 123

Defn: a |--> 7/15*a^2 + 13/5*a - 111/5

]

sage: prod(x - sigma(a) for sigma in A) == f.monic()

True

"""

try:

# this should be concordant with embeddings

return self.__embeddings[self]

except AttributeError:

self.__embeddings = {}

except KeyError:

pass

f = self.pari_polynomial('y')

# Compute the conjugates of Mod(x, f).

conj = self.pari_nf().nfgaloisconj()

# Convert these to conjugates of self.gen().

P = self._pari_absolute_structure()[1].lift()

conj = sorted([self(P(g.Mod(f))) for g in conj])

v = [self.hom([e]) for e in conj] # check=False here?

put_natural_embedding_first(v)

self.__embeddings[self] = Sequence(v, cr=(v != []), immutable=True,

check=False, universe=self.Hom(self))

return self.__embeddings[self]

def embeddings(self, K):

"""

Compute all field embeddings of self into the field K (which need

not even be a number field, e.g., it could be the complex numbers).

This will return an identical result when given K as input again.

If possible, the most natural embedding of self into K is put first

in the list.

INPUT:

- ``K`` - a number field

EXAMPLES::

sage: K.<a> = NumberField(x^3 - 2)

sage: L.<a1> = K.galois_closure(); L

Number Field in a1 with defining polynomial x^6 + 108

sage: K.embeddings(L)[0]

Ring morphism:

From: Number Field in a with defining polynomial x^3 - 2

To: Number Field in a1 with defining polynomial x^6 + 108

Defn: a |--> 1/18*a1^4

sage: K.embeddings(L) is K.embeddings(L)

True

We embed a quadratic field into a cyclotomic field::

sage: L.<a> = QuadraticField(-7)

sage: K = CyclotomicField(7)

sage: L.embeddings(K)

[

Ring morphism:

From: Number Field in a with defining polynomial x^2 + 7

To: Cyclotomic Field of order 7 and degree 6

Defn: a |--> 2*zeta7^4 + 2*zeta7^2 + 2*zeta7 + 1,

Ring morphism:

From: Number Field in a with defining polynomial x^2 + 7

To: Cyclotomic Field of order 7 and degree 6

Defn: a |--> -2*zeta7^4 - 2*zeta7^2 - 2*zeta7 - 1

]

We embed a cubic field in the complex numbers::

sage: K.<a> = NumberField(x^3 - 2)

sage: K.embeddings(CC)

[

Ring morphism:

From: Number Field in a with defining polynomial x^3 - 2

To: Complex Field with 53 bits of precision

Defn: a |--> -0.62996052494743... - 1.09112363597172*I,

Ring morphism:

From: Number Field in a with defining polynomial x^3 - 2

To: Complex Field with 53 bits of precision

Defn: a |--> -0.62996052494743... + 1.09112363597172*I,

Ring morphism:

From: Number Field in a with defining polynomial x^3 - 2

To: Complex Field with 53 bits of precision

Defn: a |--> 1.25992104989487

]

Test that :trac:`15053` is fixed::

sage: K = NumberField(x^3 - 2, 'a')

sage: K.embeddings(GF(3))

[]

"""

try:

# this should be concordant with automorphisms

return self.__embeddings[K]

except AttributeError:

self.__embeddings = {}

except KeyError:

pass

if K is self:

return self.automorphisms()

if K.characteristic() != 0:

return Sequence([], immutable=True, check=False, universe=self.Hom(K))

f = self.defining_polynomial()

r = sorted(f.roots(K, multiplicities=False))

v = [self.hom([e], check=False) for e in r]

# If there is an embedding that preserves variable names

# then it is most natural, so we put it first.

put_natural_embedding_first(v)

self.__embeddings[K] = Sequence(v, cr=v!=[], immutable=True,

check=False, universe=self.Hom(K))

return self.__embeddings[K]

def minkowski_embedding(self, B=None, prec=None):

r"""

Return an nxn matrix over RDF whose columns are the images of the

basis `\{1, \alpha, \dots, \alpha^{n-1}\}` of self over

`\QQ` (as vector spaces), where here

`\alpha` is the generator of self over

`\QQ`, i.e. self.gen(0). If B is not None, return

the images of the vectors in B as the columns instead. If prec is

not None, use RealField(prec) instead of RDF.

This embedding is the so-called "Minkowski embedding" of a number

field in `\RR^n`: given the `n` embeddings

`\sigma_1, \dots, \sigma_n` of self in

`\CC`, write `\sigma_1, \dots, \sigma_r`

for the real embeddings, and

`\sigma_{r+1}, \dots, \sigma_{r+s}` for choices of one of

each pair of complex conjugate embeddings (in our case, we simply

choose the one where the image of `\alpha` has positive

real part). Here `(r,s)` is the signature of self. Then the

Minkowski embedding is given by

.. MATH::

x \mapsto ( \sigma_1(x), \dots,

\sigma_r(x), \sqrt{2}\Re(\sigma_{r+1}(x)),

\sqrt{2}\Im(\sigma_{r+1}(x)), \dots,

\sqrt{2}\Re(\sigma_{r+s}(x)),

\sqrt{2}\Im(\sigma_{r+s}(x)))

Equivalently, this is an embedding of self in `\RR^n` so

that the usual norm on `\RR^n` coincides with

`|x| = \sum_i |\sigma_i(x)|^2` on self.

.. TODO::

This could be much improved by implementing homomorphisms

over VectorSpaces.

EXAMPLES::

sage: F.<alpha> = NumberField(x^3+2)

sage: F.minkowski_embedding()

[ 1.00000000000000 -1.25992104989487 1.58740105196820]

[ 1.41421356237... 0.8908987181... -1.12246204830...]

[0.000000000000000 1.54308184421... 1.94416129723...]

sage: F.minkowski_embedding([1, alpha+2, alpha^2-alpha])

[ 1.00000000000000 0.740078950105127 2.84732210186307]

[ 1.41421356237... 3.7193258428... -2.01336076644...]

[0.000000000000000 1.54308184421... 0.40107945302...]

sage: F.minkowski_embedding() * (alpha + 2).vector().column()

[0.740078950105127]

[ 3.7193258428...]

[ 1.54308184421...]

TESTS::

sage: emb = F.Minkowski_embedding()

doctest:warning...:

DeprecationWarning: Minkowski_embedding is deprecated. Please use minkowski_embedding instead.

See http://trac.sagemath.org/23685 for details.

"""

n = self.degree()

if prec is None:

R = sage.rings.real_double.RDF

else:

R = sage.rings.real_mpfr.RealField(prec)

r,s = self.signature()

places = self.places(prec=prec)

if B is None:

B = [ (self.gen(0))**i for i in range(n) ]

A = ZZ['x']

f = A.gen(0)**2-2

sqrt2 = f.roots(R)[1][0]

d = {}

for col in range(n):

for row in range(r):

d[(row,col)] = places[row](B[col])

for i in range(s):

z = places[r+i](B[col])

d[(r+2*i,col)] = z.real()*sqrt2

d[(r+2*i+1,col)] = z.imag()*sqrt2

M = sage.matrix.all.matrix(d)

return M

Minkowski_embedding = deprecated_function_alias(23685, minkowski_embedding)

def places(self, all_complex=False, prec=None):

"""

Return the collection of all infinite places of self.

By default, this returns the set of real places as

homomorphisms into RIF first, followed by a choice of one of

each pair of complex conjugate homomorphisms into CIF.

On the other hand, if prec is not None, we simply return places

into RealField(prec) and ComplexField(prec) (or RDF, CDF if

prec=53). One can also use ``prec=infinity``, which returns embeddings

into the field `\overline{\QQ}` of algebraic numbers (or its subfield

`\mathbb{A}` of algebraic reals); this permits exact computation, but

can be extremely slow.

There is an optional flag all_complex, which defaults to False. If

all_complex is True, then the real embeddings are returned as

embeddings into CIF instead of RIF.

EXAMPLES::

sage: F.<alpha> = NumberField(x^3-100*x+1) ; F.places()

[Ring morphism:

From: Number Field in alpha with defining polynomial x^3 - 100*x + 1

To: Real Field with 106 bits of precision

Defn: alpha |--> -10.00499625499181184573367219280,

Ring morphism:

From: Number Field in alpha with defining polynomial x^3 - 100*x + 1

To: Real Field with 106 bits of precision

Defn: alpha |--> 0.01000001000003000012000055000273,

Ring morphism:

From: Number Field in alpha with defining polynomial x^3 - 100*x + 1

To: Real Field with 106 bits of precision

Defn: alpha |--> 9.994996244991781845613530439509]

sage: F.<alpha> = NumberField(x^3+7) ; F.places()

[Ring morphism:

From: Number Field in alpha with defining polynomial x^3 + 7

To: Real Field with 106 bits of precision

Defn: alpha |--> -1.912931182772389101199116839549,

Ring morphism:

From: Number Field in alpha with defining polynomial x^3 + 7

To: Complex Field with 53 bits of precision

Defn: alpha |--> 0.956465591386195 + 1.65664699997230*I]

sage: F.<alpha> = NumberField(x^3+7) ; F.places(all_complex=True)

[Ring morphism:

From: Number Field in alpha with defining polynomial x^3 + 7

To: Complex Field with 53 bits of precision

Defn: alpha |--> -1.91293118277239,

Ring morphism:

From: Number Field in alpha with defining polynomial x^3 + 7

To: Complex Field with 53 bits of precision

Defn: alpha |--> 0.956465591386195 + 1.65664699997230*I]

sage: F.places(prec=10)

[Ring morphism:

From: Number Field in alpha with defining polynomial x^3 + 7

To: Real Field with 10 bits of precision

Defn: alpha |--> -1.9,

Ring morphism:

From: Number Field in alpha with defining polynomial x^3 + 7

To: Complex Field with 10 bits of precision

Defn: alpha |--> 0.96 + 1.7*I]

"""

if prec is None:

R = RIF

C = CIF

elif prec == 53:

R = sage.rings.real_double.RDF

C = sage.rings.complex_double.CDF

elif prec == Infinity:

R = sage.rings.all.AA

C = sage.rings.all.QQbar

else:

R = sage.rings.real_mpfr.RealField(prec)

C = sage.rings.complex_field.ComplexField(prec)

## first, find the intervals with roots, and see how much

## precision we need to approximate the roots

all_intervals = [ x[0] for x in self.defining_polynomial().roots(C) ]

## first, set up the real places

if all_complex:

real_intervals = [ x for x in all_intervals if x.imag().is_zero() ]

else:

real_intervals = [ x[0] for x in self.defining_polynomial().roots(R) ]

if prec is None:

real_places = [ self.hom([i.center()], check=False) for i in real_intervals ]

complex_places = [ self.hom([i.center()], check=False) for i in

all_intervals if i.imag() > 0 ]

else:

real_places = [ self.hom([i], check=False) for i in real_intervals ]

complex_places = [ self.hom([i], check=False) for i in

all_intervals if i.imag() > 0 ]

return real_places + complex_places

def real_places(self, prec=None):

"""

Return all real places of self as homomorphisms into RIF.

EXAMPLES::

sage: F.<alpha> = NumberField(x^4-7) ; F.real_places()

[Ring morphism:

From: Number Field in alpha with defining polynomial x^4 - 7

To: Real Field with 106 bits of precision

Defn: alpha |--> -1.626576561697785743211232345494,

Ring morphism:

From: Number Field in alpha with defining polynomial x^4 - 7

To: Real Field with 106 bits of precision

Defn: alpha |--> 1.626576561697785743211232345494]

"""

return self.places(prec=prec)[0:self.signature()[0]]

def abs_val(self, v, iota, prec=None):

r"""

Return the value `|\iota|_{v}`.

INPUT:

- ``v`` -- a place of ``K``, finite (a fractional ideal) or infinite (element of ``K.places(prec)``)

- ``iota`` -- an element of ``K``

- ``prec`` -- (default: None) the precision of the real field

OUTPUT:

The absolute value as a real number

EXAMPLES::

sage: K.<xi> = NumberField(x^3-3)

sage: phi_real = K.places()[0]

sage: phi_complex = K.places()[1]

sage: v_fin = tuple(K.primes_above(3))[0]

sage: K.abs_val(phi_real,xi^2)

2.08008382305190

sage: K.abs_val(phi_complex,xi^2)

4.32674871092223

sage: K.abs_val(v_fin,xi^2)

0.111111111111111

"""

if prec is None:

prec = 53

R = sage.rings.real_mpfr.RealField(prec)

try:

p = v.smallest_integer()

iota_ideal = self.ideal(self(iota))

exponent = - v.residue_class_degree() * iota_ideal.valuation(v)

return R(p**exponent)

except AttributeError:

if is_real_place(v):

return R(v(iota).abs())

else:

return R(v(iota).abs()**2)

def relativize(self, alpha, names, structure=None):

r"""

Given an element in self or an embedding of a subfield into self,

return a relative number field `K` isomorphic to self that is relative

over the absolute field `\QQ(\alpha)` or the domain of `alpha`, along

with isomorphisms from `K` to self and from self to `K`.

INPUT:

- ``alpha`` - an element of self or an embedding of a subfield into

self

- ``names`` - 2-tuple of names of generator for output field K and the

subfield QQ(alpha) names[0] generators K and names[1] QQ(alpha).

- ``structure`` -- an instance of

:class:`structure.NumberFieldStructure` or ``None`` (default:

``None``), if ``None``, then the resulting field's :meth:`structure`

will return isomorphisms from and to this field. Otherwise, the field

will be equipped with ``structure``.

OUTPUT:

K -- relative number field

Also, ``K.structure()`` returns from_K and to_K, where

from_K is an isomorphism from K to self and to_K is an isomorphism

from self to K.

EXAMPLES::

sage: K.<a> = NumberField(x^10 - 2)

sage: L.<c,d> = K.relativize(a^4 + a^2 + 2); L

Number Field in c with defining polynomial x^2 - 1/5*d^4 + 8/5*d^3 - 23/5*d^2 + 7*d - 18/5 over its base field

sage: c.absolute_minpoly()

x^10 - 2

sage: d.absolute_minpoly()

x^5 - 10*x^4 + 40*x^3 - 90*x^2 + 110*x - 58

sage: (a^4 + a^2 + 2).minpoly()

x^5 - 10*x^4 + 40*x^3 - 90*x^2 + 110*x - 58

sage: from_L, to_L = L.structure()

sage: to_L(a)

sage: to_L(a^4 + a^2 + 2)

sage: from_L(to_L(a^4 + a^2 + 2))

a^4 + a^2 + 2

The following demonstrates distinct embeddings of a subfield into a

larger field::

sage: K.<a> = NumberField(x^4 + 2*x^2 + 2)

sage: K0 = K.subfields(2)[0][0]; K0

Number Field in a0 with defining polynomial x^2 - 2*x + 2

sage: rho, tau = K0.embeddings(K)

sage: L0 = K.relativize(rho(K0.gen()), 'b'); L0

Number Field in b0 with defining polynomial x^2 - b1 + 2 over its base field

sage: L1 = K.relativize(rho, 'b'); L1

Number Field in b with defining polynomial x^2 - a0 + 2 over its base field

sage: L2 = K.relativize(tau, 'b'); L2

Number Field in b with defining polynomial x^2 + a0 over its base field

sage: L0.base_field() is K0

False

sage: L1.base_field() is K0

True

sage: L2.base_field() is K0

True

Here we see that with the different embeddings, the relative norms are

different::

sage: a0 = K0.gen()

sage: L1_into_K, K_into_L1 = L1.structure()

sage: L2_into_K, K_into_L2 = L2.structure()

sage: len(K.factor(41))

sage: w1 = -a^2 + a + 1; P = K.ideal([w1])

sage: Pp = L1.ideal(K_into_L1(w1)).ideal_below(); Pp == K0.ideal([4*a0 + 1])

True

sage: Pp == w1.norm(rho)

True

sage: w2 = a^2 + a - 1; Q = K.ideal([w2])

sage: Qq = L2.ideal(K_into_L2(w2)).ideal_below(); Qq == K0.ideal([-4*a0 + 9])

True

sage: Qq == w2.norm(tau)

True

sage: Pp == Qq

False

TESTS:

We can relativize over the whole field::

sage: K.<a> = NumberField(x^4 + 2*x^2 + 2)

sage: K.relativize(K.gen(), 'a')

Number Field in a0 with defining polynomial x - a1 over its base field

sage: K.relativize(2*K.gen(), 'a')

Number Field in a0 with defining polynomial x - 1/2*a1 over its base field

We can relativize over the prime field::

sage: L = K.relativize(K(1), 'a'); L

Number Field in a0 with defining polynomial x^4 + 2*x^2 + 2 over its base field

sage: L.base_field()

Number Field in a1 with defining polynomial x - 1

sage: L.base_field().base_field()

Rational Field

sage: L = K.relativize(K(2), 'a'); L

Number Field in a0 with defining polynomial x^4 + 2*x^2 + 2 over its base field

sage: L.base_field()

Number Field in a1 with defining polynomial x - 2

sage: L.base_field().base_field()

Rational Field

sage: L = K.relativize(K(0), 'a'); L

Number Field in a0 with defining polynomial x^4 + 2*x^2 + 2 over its base field

sage: L.base_field()

Number Field in a1 with defining polynomial x

sage: L.base_field().base_field()

Rational Field

We can relativize over morphisms returned by self.subfields()::

sage: L = NumberField(x^4 + 1, 'a')

sage: [L.relativize(h, 'c') for (f,h,i) in L.subfields()]

[Number Field in c with defining polynomial x^4 + 1 over its base field,

Number Field in c with defining polynomial x^2 - a1*x + 1 over its base field,

Number Field in c with defining polynomial x^2 - 1/2*a2 over its base field,

Number Field in c with defining polynomial x^2 - a3*x - 1 over its base field,

Number Field in c with defining polynomial x - a4 over its base field]

We can relativize over a relative field::

sage: K.<z> = CyclotomicField(16)

sage: L, L_into_K, _ = K.subfields(4)[0]; L

Number Field in z0 with defining polynomial x^4 + 16

sage: F, F_into_L, _ = L.subfields(2)[0]; F

Number Field in z0_0 with defining polynomial x^2 + 64

sage: L_over_F = L.relativize(F_into_L, 'c'); L_over_F

Number Field in c with defining polynomial x^2 - 1/2*z0_0 over its base field

sage: L_over_F_into_L, _ = L_over_F.structure()

sage: K_over_rel = K.relativize(L_into_K * L_over_F_into_L, 'a'); K_over_rel

Number Field in a with defining polynomial x^2 - 1/2*c over its base field

sage: K_over_rel.base_field() is L_over_F

True

sage: K_over_rel.structure()

(Relative number field morphism:

From: Number Field in a with defining polynomial x^2 - 1/2*c over its base field

To: Cyclotomic Field of order 16 and degree 8

Defn: a |--> z

c |--> 2*z^2

z0_0 |--> 8*z^4, Ring morphism:

From: Cyclotomic Field of order 16 and degree 8

To: Number Field in a with defining polynomial x^2 - 1/2*c over its base field

Defn: z |--> a)

We can relativize over a really large field::

sage: K.<a> = CyclotomicField(3^3*2^3)

sage: R = K.relativize(a^(3^2), 't'); R

Number Field in t0 with defining polynomial x^9 - t1 over its base field

sage: R.structure()

(Relative number field morphism:

From: Number Field in t0 with defining polynomial x^9 - t1 over its base field

To: Cyclotomic Field of order 216 and degree 72

Defn: t0 |--> a

t1 |--> a^9,

Ring morphism:

From: Cyclotomic Field of order 216 and degree 72

To: Number Field in t0 with defining polynomial x^9 - t1 over its base field

Defn: a |--> t0)

Only one name is required when a morphism is given (fixing :trac:`12005`)::

sage: R.<x> = PolynomialRing(QQ)

sage: K. = NumberField(x^2 + 1)

sage: L. = NumberField(x^4 - x^2 + 1)

sage: phi = K.hom(b^3, L)

sage: M.<r> = L.relativize(phi)

sage: M

Number Field in r with defining polynomial x^2 - i*x - 1 over its base field

sage: M.base_field()

Number Field in i with defining polynomial x^2 + 1

"""

# step 1: construct the abstract field generated by alpha.w

# step 2: make a relative extension of it.

# step 3: construct isomorphisms

from sage.all import vector, matrix

from sage.categories.map import is_Map

if is_Map(alpha):

# alpha better be a morphism with codomain self

if alpha.codomain() != self:

raise ValueError("Co-domain of morphism must be self")

L = alpha.domain()

alpha = alpha(L.gen()) # relativize over phi's domain

if L is QQ:

from sage.rings.all import polygen

f = polygen(QQ)

else:

f = L.defining_polynomial() # = alpha.minpoly()

names = normalize_names(1, names)

else:

# alpha must be an element coercible to self

alpha = self(alpha)

f = alpha.minpoly()

names = normalize_names(2, names)

L = NumberField(f, names[1])

# now we do some linear algebra to find the minpoly of self.gen() over L

L_into_self = L.hom([alpha])

extdeg = self.absolute_degree() // L.absolute_degree() # [ L : self ]

a = self.gen()

# we will find a linear relation between small powers of a over L

basis = [ a**i * b for i in range(extdeg) for b in map(L_into_self, L.power_basis()) ]

basis.append(a**extdeg) # this one makes the basis no longer a basis

mat = matrix([ b.vector() for b in basis ])

soln_space = mat.left_kernel(mat.row_space()(0))

# the solution space is one dimensional and the last entry is non-zero

# because a satisfies no smaller linear relation

assert soln_space.dimension() == 1

(reln, ) = soln_space.basis()

assert reln[-1] != 0

reln = reln * ~reln[-1]

# now we need to get those coeffs in L

coeff_mat = matrix(extdeg, f.degree(), list(reln)[:-1]) # easy way to divide into the correct lengths

coeffs_in_L = [ r*vector(L.power_basis()) for r in coeff_mat.rows() ]

# f is the minimal polynomial of a over L

f = L['x'](coeffs_in_L + [1])

# sanity check...

mp_in_self = self['x']([L_into_self(_) for _ in f.coefficients(sparse=False)])

assert mp_in_self(a) == 0

if structure is None:

from sage.rings.number_field.structure import RelativeFromAbsolute

structure = RelativeFromAbsolute(self, alpha)

if L is QQ:

return L.extension(f, names[0])

else:

return L.extension(f, names[0], structure=structure)

# Synonyms so that terminology appropriate to relative number fields

# can be applied to an absolute number field:

def absolute_degree(self):

"""

A synonym for degree.

EXAMPLES::

sage: K. = NumberField(x^2 + 1)

sage: K.absolute_degree()

"""

return self.degree()

def relative_degree(self):

"""

A synonym for degree.

EXAMPLES::

sage: K. = NumberField(x^2 + 1)

sage: K.relative_degree()

"""

return self.degree()

def absolute_polynomial(self):

"""

A synonym for polynomial.

EXAMPLES::

sage: K. = NumberField(x^2 + 1)

sage: K.absolute_polynomial()

x^2 + 1

"""

return self.polynomial()

def relative_polynomial(self):

"""

A synonym for polynomial.

EXAMPLES::

sage: K. = NumberField(x^2 + 1)

sage: K.relative_polynomial()

x^2 + 1

"""

return self.polynomial()

def absolute_vector_space(self):

"""

A synonym for vector_space.

EXAMPLES::

sage: K. = NumberField(x^2 + 1)

sage: K.absolute_vector_space()

(Vector space of dimension 2 over Rational Field,

Isomorphism map:

From: Vector space of dimension 2 over Rational Field

To: Number Field in i with defining polynomial x^2 + 1,

Isomorphism map:

From: Number Field in i with defining polynomial x^2 + 1

To: Vector space of dimension 2 over Rational Field)

"""

return self.vector_space()

def relative_vector_space(self):

"""

A synonym for vector_space.

EXAMPLES::

sage: K. = NumberField(x^2 + 1)

sage: K.relative_vector_space()

(Vector space of dimension 2 over Rational Field,

Isomorphism map:

From: Vector space of dimension 2 over Rational Field

To: Number Field in i with defining polynomial x^2 + 1,

Isomorphism map:

From: Number Field in i with defining polynomial x^2 + 1

To: Vector space of dimension 2 over Rational Field)

"""

return self.vector_space()

def absolute_discriminant(self):

"""

A synonym for discriminant.

EXAMPLES::

sage: K. = NumberField(x^2 + 1)

sage: K.absolute_discriminant()

-4

"""

return self.discriminant()

def relative_discriminant(self):

"""

A synonym for discriminant.

EXAMPLES::

sage: K. = NumberField(x^2 + 1)

sage: K.relative_discriminant()

-4

"""

return self.discriminant()

def absolute_different(self):

"""

A synonym for different.

EXAMPLES::

sage: K. = NumberField(x^2 + 1)

sage: K.absolute_different()

Fractional ideal (2)

"""

return self.different()

def relative_different(self):

"""

A synonym for different.

EXAMPLES::

sage: K. = NumberField(x^2 + 1)

sage: K.relative_different()

Fractional ideal (2)

"""

return self.different()

def hilbert_symbol(self, a, b, P = None):

r"""

Returns the Hilbert symbol `(a,b)_P` for a prime P of self

and non-zero elements a and b of self.

If P is omitted, return the global Hilbert symbol `(a,b)` instead.

INPUT:

- ``a``, ``b`` -- elements of self

- ``P`` -- (default: None) If `P` is ``None``, compute the global

symbol. Otherwise, `P` should be either a prime ideal of self

(which may also be given as a generator or set of generators)

or a real or complex embedding.

OUTPUT:

If a or b is zero, returns 0.

If a and b are non-zero and P is specified, returns

the Hilbert symbol `(a,b)_P`, which is 1 if the equation

`a x^2 + b y^2 = 1` has a solution in the completion of

self at P, and is -1 otherwise.

If a and b are non-zero and P is unspecified, returns 1

if the equation has a solution in self and -1 otherwise.

EXAMPLES:

Some global examples::

sage: K.<a> = NumberField(x^2 - 23)

sage: K.hilbert_symbol(0, a+5)

sage: K.hilbert_symbol(a, 0)

sage: K.hilbert_symbol(-a, a+1)

sage: K.hilbert_symbol(-a, a+2)

-1

sage: K.hilbert_symbol(a, a+5)

-1

That the latter two are unsolvable should be visible in local

obstructions. For the first, this is a prime ideal above 19.

For the second, the ramified prime above 23::

sage: K.hilbert_symbol(-a, a+2, a+2)

-1

sage: K.hilbert_symbol(a, a+5, K.ideal(23).factor()[0][0])

-1

More local examples::

sage: K.hilbert_symbol(a, 0, K.ideal(5))

sage: K.hilbert_symbol(a, a+5, K.ideal(5))

sage: K.hilbert_symbol(a+1, 13, (a+6)*K.maximal_order())

-1

sage: [emb1, emb2] = K.embeddings(AA)

sage: K.hilbert_symbol(a, -1, emb1)

-1

sage: K.hilbert_symbol(a, -1, emb2)

Ideals P can be given by generators::

sage: K.<a> = NumberField(x^5 - 23)

sage: pi = 2*a^4 + 3*a^3 + 4*a^2 + 15*a + 11

sage: K.hilbert_symbol(a, a+5, pi)

sage: rho = 2*a^4 + 3*a^3 + 4*a^2 + 15*a + 11

sage: K.hilbert_symbol(a, a+5, rho)

This also works for non-principal ideals::

sage: K.<a> = QuadraticField(-5)

sage: P = K.ideal(3).factor()[0][0]

sage: P.gens_reduced() # random, could be the other factor

(3, a + 1)

sage: K.hilbert_symbol(a, a+3, P)

sage: K.hilbert_symbol(a, a+3, [3, a+1])

Primes above 2::

sage: K.<a> = NumberField(x^5 - 23)

sage: O = K.maximal_order()

sage: p = [p[0] for p in (2*O).factor() if p[0].norm() == 16][0]

sage: K.hilbert_symbol(a, a+5, p)

sage: K.hilbert_symbol(a, 2, p)

sage: K.hilbert_symbol(-1, a-2, p)

-1

Various real fields are allowed::

sage: K.<a> = NumberField(x^3+x+1)

sage: K.hilbert_symbol(a/3, 1/2, K.embeddings(RDF)[0])

sage: K.hilbert_symbol(a/5, -1, K.embeddings(RR)[0])

-1

sage: [K.hilbert_symbol(a, -1, e) for e in K.embeddings(AA)]

[-1]

Real embeddings are not allowed to be disguised as complex embeddings::

sage: K.<a> = QuadraticField(5)

sage: K.hilbert_symbol(-1, -1, K.embeddings(CC)[0])

Traceback (most recent call last):

...

ValueError: Possibly real place (=Ring morphism:

From: Number Field in a with defining polynomial x^2 - 5

To: Complex Field with 53 bits of precision

Defn: a |--> -2.23606797749979) given as complex embedding in hilbert_symbol. Is it real or complex?

sage: K.hilbert_symbol(-1, -1, K.embeddings(QQbar)[0])

Traceback (most recent call last):

...

ValueError: Possibly real place (=Ring morphism:

From: Number Field in a with defining polynomial x^2 - 5

To: Algebraic Field

Defn: a |--> -2.236067977499790?) given as complex embedding in hilbert_symbol. Is it real or complex?

sage: K. = QuadraticField(-5)

sage: K.hilbert_symbol(-1, -1, K.embeddings(CDF)[0])

sage: K.hilbert_symbol(-1, -1, K.embeddings(QQbar)[0])

a and b do not have to be integral or coprime::

sage: K. = QuadraticField(-1)

sage: O = K.maximal_order()

sage: K.hilbert_symbol(1/2, 1/6, 3*O)

sage: p = 1+i

sage: K.hilbert_symbol(p, p, p)

sage: K.hilbert_symbol(p, 3*p, p)

-1

sage: K.hilbert_symbol(3, p, p)

-1

sage: K.hilbert_symbol(1/3, 1/5, 1+i)

sage: L = QuadraticField(5, 'a')

sage: L.hilbert_symbol(-3, -1/2, 2)

Various other examples::

sage: K.<a> = NumberField(x^3+x+1)

sage: K.hilbert_symbol(-6912, 24, -a^2-a-2)

sage: K.<a> = NumberField(x^5-23)

sage: P = K.ideal(-1105*a^4 + 1541*a^3 - 795*a^2 - 2993*a + 11853)

sage: Q = K.ideal(-7*a^4 + 13*a^3 - 13*a^2 - 2*a + 50)

sage: b = -a+5

sage: K.hilbert_symbol(a,b,P)

sage: K.hilbert_symbol(a,b,Q)

sage: K.<a> = NumberField(x^5-23)

sage: P = K.ideal(-1105*a^4 + 1541*a^3 - 795*a^2 - 2993*a + 11853)

sage: K.hilbert_symbol(a, a+5, P)

sage: K.hilbert_symbol(a, 2, P)

sage: K.hilbert_symbol(a+5, 2, P)

-1

sage: K.<a> = NumberField(x^3 - 4*x + 2)

sage: K.hilbert_symbol(2, -2, K.primes_above(2)[0])

Check that the bug reported at :trac:`16043` has been fixed::

sage: K.<a> = NumberField(x^2 + 5)

sage: p = K.primes_above(2)[0]; p

Fractional ideal (2, a + 1)

sage: K.hilbert_symbol(2*a, -1, p)

sage: K.hilbert_symbol(2*a, 2, p)

-1

sage: K.hilbert_symbol(2*a, -2, p)

-1

AUTHOR:

- Aly Deines (2010-08-19): part of the doctests

- Marco Streng (2010-12-06)

"""

if a.is_zero() or b.is_zero():

return 0

a = self(a)

b = self(b)

if P is None:

return pari(self).nfhilbert(a, b)

from sage.categories.map import Map

from sage.categories.all import Rings

if isinstance(P, Map) and P.category_for().is_subcategory(Rings()):

# P is a morphism of Rings

if P.domain() is not self:

raise ValueError("Domain of P (=%s) should be self (=%s) in self.hilbert_symbol" % (P, self))

codom = P.codomain()

from sage.rings.complex_field import is_ComplexField

from sage.rings.complex_interval_field import is_ComplexIntervalField

from sage.rings.real_mpfr import is_RealField

from sage.rings.real_mpfi import is_RealIntervalField

from sage.rings.all import (AA, CDF, QQbar, RDF)

if is_ComplexField(codom) or is_ComplexIntervalField(codom) or \

codom is CDF or codom is QQbar:

if P(self.gen()).imag() == 0:

raise ValueError("Possibly real place (=%s) given as complex embedding in hilbert_symbol. Is it real or complex?" % P)

return 1

if is_RealField(codom) or codom is RDF or codom is AA:

if P(a) > 0 or P(b) > 0:

return 1

return -1

if not is_NumberFieldIdeal(P):

P = self.ideal(P)

if P.number_field() is not self:

raise ValueError("P (=%s) should be an ideal of self (=%s) in hilbert_symbol, not of %s" % (P, self, P.number_field()))

if not P.is_prime():

raise ValueError("Non-prime ideal P (=%s) in hilbert_symbol" % P)

return pari(self).nfhilbert(a, b, P.pari_prime())

def hilbert_conductor(self,a,b):

"""

This is the product of all (finite) primes where the Hilbert symbol is -1.

What is the same, this is the (reduced) discriminant of the quaternion

algebra `(a,b)` over a number field.

INPUT:

- ``a``, ``b`` -- elements of the number field ``self``

OUTPUT:

- squarefree ideal of the ring of integers of ``self``

EXAMPLES::

sage: F.<a> = NumberField(x^2-x-1)

sage: F.hilbert_conductor(2*a,F(-1))

Fractional ideal (2)

sage: K. = NumberField(x^3-4*x+2)

sage: K.hilbert_conductor(K(2),K(-2))

Fractional ideal (1)

sage: K.hilbert_conductor(K(2*b),K(-2))

Fractional ideal (b^2 + b - 2)

AUTHOR:

- Aly Deines

"""

a, b = self(a), self(b)

d = self.ideal(1)

for p in union(union( self.ideal(2).prime_factors(), self.ideal(a).prime_factors()), self.ideal(b).prime_factors()):

if self.hilbert_symbol(a,b,p) == -1:

d *= p

return d

def elements_of_bounded_height(self,bound,precision=53,LLL=False):

r"""

Return an iterator over the elements of ``self`` with relative

multiplicative height at most ``bound``.

The algorithm requires floating point arithmetic, so the user is

allowed to specify the precision for such calculations.

It might be helpful to work with an LLL-reduced system of fundamental

units, so the user has the option to perform an LLL reduction for the

fundamental units by setting ``LLL`` to True.

Certain computations may be faster assuming GRH, which may be done

globally by using the number_field(True/False) switch.

For details: See [Doyle-Krumm]_.

INPUT:

- ``bound`` - a real number

- ``precision`` - (default: 53) a positive integer

- ``LLL`` - (default: False) a boolean value

OUTPUT:

- an iterator of number field elements

.. WARNING::

In the current implementation, the output of the algorithm cannot be

guaranteed to be correct due to the necessity of floating point

computations. In some cases, the default 53-bit precision is

considerably lower than would be required for the algorithm to

generate correct output.

.. TODO::

Should implement a version of the algorithm that guarantees correct

output. See Algorithm 4 in [Doyle-Krumm]_ for details of an

implementation that takes precision issues into account.

EXAMPLES:

There are no elements in a number field with multiplicative height less

than 1::

sage: K.<g> = NumberField(x^5 - x + 19)

sage: list(K.elements_of_bounded_height(0.9))

[]

The only elements in a number field of height 1 are 0 and the roots of

unity::

sage: K.<a> = NumberField(x^2 + x + 1)

sage: list(K.elements_of_bounded_height(1))

[0, a + 1, a, -1, -a - 1, -a, 1]

sage: K.<a> = CyclotomicField(20)

sage: len(list(K.elements_of_bounded_height(1)))

The elements in the output iterator all have relative multiplicative

height at most the input bound::

sage: K.<a> = NumberField(x^6 + 2)

sage: L = K.elements_of_bounded_height(5)

sage: for t in L:

....: exp(6*t.global_height())

....:

1.00000000000000

2.00000000000000

4.00000000000000

sage: K.<a> = NumberField(x^2 - 71)

sage: L = K.elements_of_bounded_height(20)

sage: all(exp(2*t.global_height()) <= 20 for t in L) # long time (5 s)

True

sage: K.<a> = NumberField(x^2 + 17)

sage: L = K.elements_of_bounded_height(120)

sage: len(list(L))

9047

sage: K.<a> = NumberField(x^4 - 5)

sage: L = K.elements_of_bounded_height(50)

sage: len(list(L)) # long time (2 s)

2163

sage: K.<a> = CyclotomicField(13)

sage: L = K.elements_of_bounded_height(2)

sage: len(list(L)) # long time (3 s)

sage: K.<a> = NumberField(x^6 + 2)

sage: L = K.elements_of_bounded_height(60, precision=100)

sage: len(list(L)) # long time (5 s)

1899

sage: K.<a> = NumberField(x^4 - x^3 - 3*x^2 + x + 1)

sage: L = K.elements_of_bounded_height(10, LLL=true)

sage: len(list(L))

AUTHORS:

- John Doyle (2013)

- David Krumm (2013)

"""

from sage.rings.number_field.bdd_height import bdd_height, bdd_height_iq

r1, r2 = self.signature()

r = r1 + r2 - 1

if self.degree() == 2 and r == 0:

return bdd_height_iq(self, bound)

else:

return bdd_height(self, bound, precision, LLL)

class NumberField_cyclotomic(NumberField_absolute):

"""

Create a cyclotomic extension of the rational field.

The command CyclotomicField(n) creates the n-th cyclotomic field,

obtained by adjoining an n-th root of unity to the rational field.

EXAMPLES::

sage: CyclotomicField(3)

Cyclotomic Field of order 3 and degree 2

sage: CyclotomicField(18)

Cyclotomic Field of order 18 and degree 6

sage: z = CyclotomicField(6).gen(); z

zeta6

sage: z^3

-1

sage: (1+z)^3

6*zeta6 - 3

sage: K = CyclotomicField(197)

sage: loads(K.dumps()) == K

True

sage: loads((z^2).dumps()) == z^2

True

sage: cf12 = CyclotomicField( 12 )

sage: z12 = cf12.0

sage: cf6 = CyclotomicField( 6 )

sage: z6 = cf6.0

sage: FF = Frac( cf12['x'] )

sage: x = FF.0

sage: z6*x^3/(z6 + x)

zeta12^2*x^3/(x + zeta12^2)

sage: cf6 = CyclotomicField(6) ; z6 = cf6.gen(0)

sage: cf3 = CyclotomicField(3) ; z3 = cf3.gen(0)

sage: cf3(z6)

zeta3 + 1

sage: cf6(z3)

zeta6 - 1

sage: type(cf6(z3))

sage: cf1 = CyclotomicField(1) ; z1 = cf1.0

sage: cf3(z1)

sage: type(cf3(z1))

"""

def __init__(self, n, names, embedding=None, assume_disc_small=False, maximize_at_primes=None):

"""

A cyclotomic field, i.e., a field obtained by adjoining an n-th

root of unity to the rational numbers.

EXAMPLES::

sage: k = CyclotomicField(3)

sage: type(k)

TESTS:

The ``gcd`` and ``xgcd`` methods do not agree on this field, see

:trac:`23274`::

sage: TestSuite(k).run()

Failure in _test_gcd_vs_xgcd:

...

AssertionError: The methods gcd and xgcd disagree on Cyclotomic Field of order 3 and degree 2:

gcd(0,2) = 1

xgcd(0,2) = (2, 0, 1)

------------------------------------------------------------

The following tests failed: _test_gcd_vs_xgcd

sage: type(CyclotomicField(4).zero())

sage: type(CyclotomicField(6).one())

sage: type(CyclotomicField(6).an_element())

sage: type(CyclotomicField(15).zero())

"""

f = QQ['x'].cyclotomic_polynomial(n)

if names[0].startswith('zeta'):

latex_name = "\\zeta_{%s}"%n

else:

latex_name = latex_variable_name(names[0])

self.__n = n = Integer(n)

NumberField_absolute.__init__(self, f,

name= names,

latex_name=latex_name,

check=False,

embedding = embedding,

assume_disc_small=assume_disc_small,

maximize_at_primes=maximize_at_primes)

if n%2:

self.__zeta_order = 2*n

else:

self.__zeta_order = n

## quadratic number fields require this:

if f.degree() == 2:

# define a boolean flag as for NumberField_quadratic to know, which

# square root we choose (True means no embedding or positive

# imaginary value).

# Note that the test is done with NumberFieldElement and not with

# NumberFieldElement_quadratic which requires somehow this flag.

# As a consequence, a result of _an_element_() with the wrong class

# is cached during the call to has_coerce_map_from. We reset the

# cache afterwards.

self._standard_embedding = not CDF.has_coerce_map_from(self) or CDF(self.gen()).imag() > 0

self._cache_an_element = None

self._element_class = number_field_element_quadratic.NumberFieldElement_quadratic

if n == 4:

self._D = ZZ(-1)

self._NumberField_generic__gen = self._element_class(self, (QQ(0), QQ(1)))

else:

## n is 3 or 6

self._D = ZZ(-3)

one_half = ZZ(1)/ZZ(2)

if n == 3:

self._NumberField_generic__gen = self._element_class(self, (one_half-1, one_half))

else:

self._NumberField_generic__gen = self._element_class(self, (one_half, one_half))

# NumberField_absolute.__init__(...) set _zero_element and

# _one_element to NumberFieldElement_absolute values, which is

# wrong (and dangerous; such elements can actually be used to

# crash Sage: see #5316). Overwrite them with correct values.

self._zero_element = self._element_class(self, (QQ(0),QQ(0)))

self._one_element = self._element_class(self, (QQ(1),QQ(0)))

zeta = self.gen()

zeta._set_multiplicative_order(n)

self._init_embedding_approx()

def construction(self):

"""

Return data defining a functorial construction of ``self``.

EXAMPLES::

sage: F, R = CyclotomicField(5).construction()

sage: R

Rational Field

sage: F.polys

[x^4 + x^3 + x^2 + x + 1]

sage: F.names

['zeta5']

sage: F.embeddings

[0.309016994374948? + 0.951056516295154?*I]

sage: F.structures

[None]

"""

F,R = NumberField_generic.construction(self)

F.cyclotomic = self.__n

return F,R

def _magma_init_(self, magma):

"""

Function returning a string to create this cyclotomic field in

Magma.

.. note::

The Magma generator name is also initialized to be the same

as for the Sage field.

EXAMPLES::

sage: K=CyclotomicField(7,'z')

sage: K._magma_init_(magma) # optional - magma

'SageCreateWithNames(CyclotomicField(7),["z"])'

sage: K=CyclotomicField(7,'zeta')

sage: K._magma_init_(magma) # optional - magma

'SageCreateWithNames(CyclotomicField(7),["zeta"])'

"""

s = 'CyclotomicField(%s)'%self.__n

return magma._with_names(s, self.variable_names())

def _gap_init_(self):

"""

Return a string that provides a representation of ``self`` in GAP.

TESTS::

sage: K = CyclotomicField(8)

sage: gap(K) # indirect doctest

CF(8)

sage: gap(K.0)

E(8)

sage: K(gap(K.0^5)); K(gap(K.0^5))==K.0^5

-zeta8

True

The following was the motivating example to introduce

a genuine representation of cyclotomic fields in the

GAP interface -- see :trac:`5618`. ::

sage: H = AlternatingGroup(4)

sage: g = H.list()[1]

sage: K = H.subgroup([g])

sage: z = CyclotomicField(3).an_element(); z

zeta3

sage: c = K.character([1,z,z**2]); c

Character of Subgroup of (Alternating group of order 4!/2 as a permutation group) generated by [(1,2,3)]

sage: c(g^2); z^2

-zeta3 - 1

"""

return 'CyclotomicField(%s)'%self.__n

def _libgap_(self):

"""

Return a LibGAP representation of ``self``.

TESTS::

sage: K = CyclotomicField(8)

sage: K._libgap_()

CF(8)

sage: libgap(K) # indirect doctest

CF(8)

"""

from sage.libs.gap.libgap import libgap

return libgap.CyclotomicField(self.__n)

def _repr_(self):

r"""

Return string representation of this cyclotomic field.

The "order" of the cyclotomic field `\QQ(\zeta_n)`

in the string output refers to the order of the `\zeta_n`,

i.e., it is the integer `n`. The degree is the degree of

the field as an extension of `\QQ`.

EXAMPLES::

sage: CyclotomicField(4)._repr_()

'Cyclotomic Field of order 4 and degree 2'

sage: CyclotomicField(400)._repr_()

'Cyclotomic Field of order 400 and degree 160'

"""

return "Cyclotomic Field of order %s and degree %s"%(

self.__n, self.degree())

def _n(self):

"""

Return the n used to create this cyclotomic field.

EXAMPLES::

sage: CyclotomicField(3).zeta_order()

sage: CyclotomicField(3)._n()

"""

return self.__n

def _latex_(self):

"""

Return the latex representation of this cyclotomic field.

EXAMPLES::

sage: Z = CyclotomicField(4)

sage: Z.gen()

zeta4

sage: latex(Z) # indirect doctest

\Bold{Q}(\zeta_{4})

Latex printing respects the generator name::

sage: k.<a> = CyclotomicField(4)

sage: latex(k)

\Bold{Q}[a]/(a^{2} + 1)

sage: k

Cyclotomic Field of order 4 and degree 2

sage: k.gen()

TESTS:

We check that the bug reported on :trac:`8938` is fixed::

sage: C5.<z> = CyclotomicField(5)

sage: P.<s, t> = C5[]

sage: f = (z^2 + z)*s

sage: f

(z^2 + z)*s

sage: latex(f)

\left(z^{2} + z\right) s

"""

v = self.latex_variable_name()

if v.startswith('\\zeta_'):

return "%s(%s)"%(latex(QQ), v)

else:

return NumberField_generic._latex_(self)

def _coerce_map_from_(self, K):

r"""

Return a coercion map from `K` to ``self``, or None.

The cyclotomic field `\QQ(\zeta_n)` coerces into the

cyclotomic field `\QQ(\zeta_m)` if and only if `n' \mid m`,

where `n'` is the odd part of `n` if `4 \nmid n` and `n' = n`

otherwise.

The morphism is consistent with the chosen embedding into `\CC`.

If `K` is not a cyclotomic field, the normal coercion rules

for number fields are used.

EXAMPLES::

sage: K.<a> = CyclotomicField(12)

sage: L. = CyclotomicField(132)

sage: L.coerce_map_from(K) # indirect doctest

Generic morphism:

From: Cyclotomic Field of order 12 and degree 4

To: Cyclotomic Field of order 132 and degree 40

Defn: a -> b^11

sage: a + b

b^11 + b

sage: L.coerce_map_from(CyclotomicField(4, 'z'))

Generic morphism:

From: Cyclotomic Field of order 4 and degree 2

To: Cyclotomic Field of order 132 and degree 40

Defn: z -> b^33

sage: L.coerce_map_from(CyclotomicField(5, 'z')) is None

True

sage: K.<a> = CyclotomicField(3)

sage: L. = CyclotomicField(6)

sage: L.coerce_map_from(K)

Generic morphism:

From: Cyclotomic Field of order 3 and degree 2

To: Cyclotomic Field of order 6 and degree 2

Defn: a -> b - 1

sage: K.coerce_map_from(L)

Generic morphism:

From: Cyclotomic Field of order 6 and degree 2

To: Cyclotomic Field of order 3 and degree 2

Defn: b -> a + 1

sage: CyclotomicField(33).coerce_map_from(CyclotomicField(66))

Generic morphism:

From: Cyclotomic Field of order 66 and degree 20

To: Cyclotomic Field of order 33 and degree 20

Defn: zeta66 -> -zeta33^17

sage: CyclotomicField(15).coerce_map_from(CyclotomicField(6))

Generic morphism:

From: Cyclotomic Field of order 6 and degree 2

To: Cyclotomic Field of order 15 and degree 8

Defn: zeta6 -> zeta15^5 + 1

Check that :trac:`12632` is fixed::

sage: K1 = CyclotomicField(1); K2 = CyclotomicField(2)

sage: K1.coerce_map_from(K2)

Generic morphism:

From: Cyclotomic Field of order 2 and degree 1

To: Cyclotomic Field of order 1 and degree 1

Defn: zeta2 -> -1

Check that custom embeddings are respected (:trac:`13765`)::

sage: z105 = CDF(exp(2*pi*I/105))

sage: Ka.<a> = CyclotomicField(105, embedding=z105^11)

sage: Kb. = CyclotomicField(35, embedding=z105^6)

sage: Ka.coerce_map_from(Kb)

Generic morphism:

From: Cyclotomic Field of order 35 and degree 24

To: Cyclotomic Field of order 105 and degree 48

Defn: b -> -a^44 - a^42 + a^39 + a^37 + a^35 - a^29 - a^27 - a^25 + a^24 - a^23 + a^22 - a^21 + a^20 + a^18 + a^16 - a^12 - a^10 - a^8 - a^6 + a^5 + a^3 + a

sage: CC(b)

0.936234870639737 + 0.351374824081343*I

sage: CC(-a^44 - a^42 + a^39 + a^37 + a^35 - a^29 - a^27 - a^25 + a^24 - a^23 + a^22 - a^21 + a^20 + a^18 + a^16 - a^12 - a^10 - a^8 - a^6 + a^5 + a^3 + a)

0.936234870639731 + 0.351374824081341*I

sage: z15 = CDF(exp(2*pi*I/15))

sage: CyclotomicField(15).coerce_map_from(CyclotomicField(6, embedding=-z15^5))

Generic morphism:

From: Cyclotomic Field of order 6 and degree 2

To: Cyclotomic Field of order 15 and degree 8

Defn: zeta6 -> -zeta15^5

sage: CyclotomicField(15, embedding=z15^4).coerce_map_from(CyclotomicField(6, embedding=-z15^5))

Generic morphism:

From: Cyclotomic Field of order 6 and degree 2

To: Cyclotomic Field of order 15 and degree 8

Defn: zeta6 -> -zeta15^5

Check transitivity of coercion embeddings (:trac:`20513`)::

sage: K60.<zeta60> = CyclotomicField(60)

sage: K30.<zeta30> = CyclotomicField(30, embedding=zeta60**14)

sage: K15.<zeta15> = CyclotomicField(15, embedding=zeta30**26)

sage: K5.<zeta5> = CyclotomicField(5, embedding=zeta15**12)

sage: K60.has_coerce_map_from(K5)

True

sage: K60(zeta5)

-zeta60^14 - zeta60^12 + zeta60^6 + zeta60^4 - 1

sage: _ == zeta60**(14*26*12)

True

"""

if isinstance(K, NumberField_cyclotomic):

if (self.coerce_embedding() is None or K.coerce_embedding() is None):

return None

ambient_field = self.coerce_embedding().codomain()

if not ambient_field.has_coerce_map_from(K.coerce_embedding().codomain()):

return None

Kn = K.__n

n = self.__n

if Kn.divides(n):

return number_field_morphisms.CyclotomicFieldEmbedding(K, self)

if Kn == 2 and n == 1:

# see #12632

return number_field_morphisms.NumberFieldEmbedding(K, self, -self.gen())

if Kn % 4 == 2 and (Kn//2).divides(n):

e = self._log_gen(ambient_field(-K.gen()))

return number_field_morphisms.NumberFieldEmbedding(K, self, -self.gen() ** e)

else:

return None

elif self.degree() == 2:

if K is ZZ:

return number_field_element_quadratic.Z_to_quadratic_field_element(self)

if K is QQ:

return number_field_element_quadratic.Q_to_quadratic_field_element(self)

return NumberField_absolute._coerce_map_from_(self, K)

def _log_gen(self, x):

"""

Returns an integer `e` such that `self.gen()^e == x`, or `None`

if no such integer exists. This is primarily used to construct

embedding-respecting coercions.

If `x` is complex, the result is either an integer `e` such

that the absolute value of `self.gen()^e-x` is small or

`None` if no such `e` is found.

EXAMPLES::

sage: K.<a> = CyclotomicField(5)

sage: K._log_gen(CDF(a))

sage: K._log_gen(CDF(a^4))

sage: zeta105 = CC(exp(2*pi*i/105))

sage: K.<a> = CyclotomicField(105, embedding=zeta105^13)

sage: zeta105^13, CC(a)

(0.712376096951345 + 0.701797902883992*I, 0.712376096951345 + 0.701797902883991*I)

sage: K._log_gen(zeta105^26)

sage: K._log_gen(zeta105)

sage: zeta105, CC(a^97)

(0.998210129767735 + 0.0598041539450342*I, 0.998210129767736 + 0.0598041539450313*I)

sage: K._log_gen(zeta105^3)

sage: zeta105^3, CC(a)^81

(0.983929588598630 + 0.178556894798637*I, 0.983929588598631 + 0.178556894798635*I)

sage: K.<a> = CyclotomicField(5, embedding=None)

sage: K._log_gen(CDF(.5, -.8)) is None

True

sage: zeta5 = cyclotomic_polynomial(5).change_ring(Qp(11)).roots()[0][0]

sage: zeta5 ^ 5

1 + O(11^20)

sage: K.<a> = CyclotomicField(5, embedding=zeta5^2)

sage: K._log_gen(zeta5)

sage: K60.<zeta60> = CyclotomicField(60)

sage: K30.<zeta30> = CyclotomicField(30, embedding=zeta60**2)

sage: K15.<zeta15> = CyclotomicField(15, embedding=zeta30**2)

sage: K5.<zeta5> = CyclotomicField(5, embedding=zeta15**12)

sage: K60._log_gen(zeta30)

sage: K60._log_gen(zeta15)

sage: K60._log_gen(zeta5)

sage: K5._log_gen(zeta15**3)

"""

X = x.parent()

gen = self.gen()

if self.has_coerce_map_from(X):

Y = self

x = self(x)

elif X.has_coerce_map_from(self):

Y = X

gen = X(self.gen())

else:

return

n = self._n()

if CDF.has_coerce_map_from(Y):

x = CDF(x)

gen = CDF(gen)

# Let zeta = e^(2*pi*i/n)

two_pi = 2*RDF.pi()

a = (n * x.arg() / two_pi).round() # x = zeta^a

b = (n * gen.arg() / two_pi).round() # gen = zeta^b

e = mod(a/b, n).lift() # e is the expected result

if abs(gen**e-x) < 1/n: # a sanity check

return e

else:

# NOTE: this can be *very* slow!

gen_pow_e = 1

for e in range(n):

if gen_pow_e == x:

return e

gen_pow_e *= gen

def _element_constructor_(self, x, check=True):

"""

Create an element of this cyclotomic field from `x`.

EXAMPLES:

The following example illustrates coercion from the

cyclotomic field Q(zeta_42) to the cyclotomic field Q(zeta_6), in

a case where such coercion is defined::

sage: k42 = CyclotomicField(42)

sage: k6 = CyclotomicField(6)

sage: a = k42.gen(0)

sage: b = a^7

sage: b

zeta42^7

sage: k6(b) # indirect doctest

zeta6

sage: b^2

zeta42^7 - 1

sage: k6(b^2)

zeta6 - 1

Conversion of elements of the :class:`~sage.rings.universal_cyclotomic_field.UniversalCyclotomicField`::

sage: CF = CyclotomicField(5)

sage: UCF.<E> = UniversalCyclotomicField()

sage: CF(E(5))

zeta5

sage: CF = CyclotomicField(10)

sage: CF(E(5))

zeta10^2

Coercion of GAP cyclotomic elements is also supported::

sage: CyclotomicField(18)(gap('E(3)')) # indirect doctest

zeta18^3 - 1

Converting from rings of integers::

sage: K.<z> = CyclotomicField(7)

sage: O = K.maximal_order()

sage: K(O.1)

sage: K(O.1^2 + O.1 - 2)

z^2 + z - 2

"""

if isinstance(x, number_field_element.NumberFieldElement):

if isinstance(x.parent(), NumberField_cyclotomic):

return self._coerce_from_other_cyclotomic_field(x)

else:

return NumberField_absolute._element_constructor_(self, x)

elif isinstance(x, pari_gen):

return NumberField_absolute._element_constructor_(self, x, check=check)

elif (sage.interfaces.gap.is_GapElement(x) or

isinstance(x, sage.libs.gap.element.GapElement)):

return self._coerce_from_gap(x)

elif isinstance(x,str):

return self._convert_from_str(x)

# late import because of speed

from sage.rings.universal_cyclotomic_field import UniversalCyclotomicFieldElement

if isinstance(x,UniversalCyclotomicFieldElement):

return x.to_cyclotomic_field(self)

else:

return self._convert_non_number_field_element(x)

# TODO:

# The following is very nice and much more flexible / powerful.

# However, it is simply not *consistent*, since it totally

# breaks the doctests in eisenstein_submodule.py.

# FIX THIS.

## def _will_be_better_coerce_from_other_cyclotomic_field(self, x, only_canonical=False):

## """

## Coerce an element x of a cyclotomic field into self, if at all possible.

## INPUT:

## x -- number field element

## only_canonical -- bool (default: False); Attempt to work,

## even in some cases when x is not in a subfield of

## the cyclotomics (as long as x is a root of unity).

## EXAMPLES::

## sage: k5 = CyclotomicField(5)

## sage: k3 = CyclotomicField(3)

## sage: k15 = CyclotomicField(15)

## sage: k15._coerce_from_other_cyclotomic_field(k3.gen())

## zeta15^5

## sage: k15._coerce_from_other_cyclotomic_field(k3.gen()^2 + 17/3)

## -zeta15^5 + 14/3

## sage: k3._coerce_from_other_cyclotomic_field(k15.gen()^5)

## zeta3

## sage: k3._coerce_from_other_cyclotomic_field(-2/3 * k15.gen()^5 + 2/3)

## -2/3*zeta3 + 2/3

## """

## K = x.parent()

## if K is self:

## return x

## n = K.zeta_order()

## m = self.zeta_order()

## self_gen = self.gen()

## if m % n == 0: # easy case

## # pass this off to a method in the element class

## # it can be done very quickly and easily by the cython<->NTL

## # interface there

## return x._lift_cyclotomic_element(self)

## # Whatever happens below, it has to be consistent with

## # zeta_r |---> (zeta_s)^m

## if m % 2 and not n%2:

## m *= 2

## self_gen = -self_gen

## if only_canonical and m % n:

## raise TypeError, "no canonical coercion"

## if not is_CyclotomicField(K):

## raise TypeError, "x must be in a cyclotomic field"

## v = x.list()

## # Find the smallest power r >= 1 of the generator g of K that is in self,

## # i.e., find the smallest r such that g^r has order dividing m.

## d = sage.arith.all.gcd(m,n)

## r = n // d

## # Since we use the power basis for cyclotomic fields, if every

## # v[i] with i not divisible by r is 0, then we're good.

## # If h generates self and has order m, then the element g^r

## # maps to the power of self of order gcd(m,n)., i.e., h^(m/gcd(m,n))

## #

## z = self_gen**(m // d)

## w = self(1)

## a = self(0)

## for i in range(len(v)):

## if i%r:

## if v[i]:

## raise TypeError, "element does not belong to cyclotomic field"

## else:

## a += w*v[i]

## w *= z

## return a

def _coerce_from_other_cyclotomic_field(self, x, only_canonical=False):

"""

Coerce an element x of a cyclotomic field into self, if at all

possible.

INPUT:

- ``x`` - number field element

- ``only_canonical`` - bool (default: False); Attempt

to work, even in some cases when x is not in a subfield of the

cyclotomics (as long as x is a root of unity).

EXAMPLES::

sage: K = CyclotomicField(24) ; L = CyclotomicField(48)

sage: L._coerce_from_other_cyclotomic_field(K.0+1)

zeta48^2 + 1

sage: K(L.0**2)

zeta24

"""

K = x.parent()

if K is self:

return x

n = K._n()

m = self._n()

if m % n == 0: # easy case

# pass this off to a method in the element class

# it can be done very quickly and easily by the

# Cython<->NTL interface there

return x._lift_cyclotomic_element(self)

else:

if only_canonical:

raise TypeError

n = x.multiplicative_order()

m = self.zeta_order()

if m % n == 0:

# Harder case. E.g., x = (zeta_42)^7 and

# self.__zeta = zeta_6, so it is possible to

# coerce x in, but not zeta_42 in.

# Algorithm:

# 1. Compute self.__zeta as an element

# of K = parent of x. Call this y.

# 2. Write x as a power r of y.

# TODO: we do step two STUPIDLY.

# 3. Return self.__zeta to the power r.

y = K(self.zeta(m))

z = y

for r in range(y.multiplicative_order()):

if z == x:

return self.zeta(m)**(r+1)

z *= y

raise TypeError("Cannot coerce %s into %s"%(x,self))

return self._element_class(self, g)

def _coerce_from_gap(self, x):

"""

Attempt to coerce a GAP number field element into this cyclotomic

field.

EXAMPLES::

sage: k5.<z> = CyclotomicField(5)

sage: gap('E(5)^7 + 3')

-3*E(5)-2*E(5)^2-3*E(5)^3-3*E(5)^4

sage: w = gap('E(5)^7 + 3')

sage: z^7 + 3

z^2 + 3

sage: k5(w) # indirect doctest

z^2 + 3

It may be that GAP uses a name for the generator of the cyclotomic field.

We can deal with this case, if this name coincides with the name in Sage::

sage: F = CyclotomicField(8)

sage: z = F.gen()

sage: a = gap(z+1/z); a

E(8)-E(8)^3

sage: F(a)

-zeta8^3 + zeta8

Matrices over cyclotomic fields are correctly dealt with it as well::

sage: b = gap(Matrix(F,[[z^2,1],[0,a+1]])); b

[ [ E(4), 1 ], [ 0, 1+E(8)-E(8)^3 ] ]

sage: b[1,2]

sage: F(b[1,2])

sage: matrix(F, b)

[ zeta8^2 1]

[ 0 -zeta8^3 + zeta8 + 1]

It also word with libGAP instead of GAP::

sage: b = libgap.eval('[[E(4), 1], [0, 1+E(8)-E(8)^3]]')

sage: matrix(F, b)

[ zeta8^2 1]

[ 0 -zeta8^3 + zeta8 + 1]

"""

if x.IsRat():

return self(QQ(x))

coeffs = x.CoeffsCyc(self.__n)

zeta = self.gen()

return sum(QQ(c)*zeta**i for i,c in enumerate(coeffs))

def _Hom_(self, codomain, cat=None):

"""

Return homset of homomorphisms from the cyclotomic field self to

the number field codomain.

The cat option is currently ignored.

EXAMPLES:

This function is implicitly called by the Hom method or

function.

sage: K.<a> = NumberField(x^2 + 3); K

Number Field in a with defining polynomial x^2 + 3

sage: CyclotomicField(3).Hom(K) # indirect doctest

Set of field embeddings from Cyclotomic Field of order 3 and degree 2 to Number Field in a with defining polynomial x^2 + 3

sage: End(CyclotomicField(21))

Automorphism group of Cyclotomic Field of order 21 and degree 12

"""

if is_NumberFieldHomsetCodomain(codomain):

from . import morphism

return morphism.CyclotomicFieldHomset(self, codomain)

else:

raise TypeError

def is_galois(self):

"""

Return True since all cyclotomic fields are automatically Galois.

EXAMPLES::

sage: CyclotomicField(29).is_galois()

True

"""

return True

def is_isomorphic(self, other):

"""

Return True if the cyclotomic field self is isomorphic as a number

field to other.

EXAMPLES::

sage: CyclotomicField(11).is_isomorphic(CyclotomicField(22))

True

sage: CyclotomicField(11).is_isomorphic(CyclotomicField(23))

False

sage: CyclotomicField(3).is_isomorphic(NumberField(x^2 + x +1, 'a'))

True

sage: CyclotomicField(18).is_isomorphic(CyclotomicField(9))

True

sage: CyclotomicField(10).is_isomorphic(NumberField(x^4 - x^3 + x^2 - x + 1, 'b'))

True

Check :trac:`14300`::

sage: K = CyclotomicField(4)

sage: N = K.extension(x^2-5, 'z')

sage: K.is_isomorphic(N)

False

sage: K.is_isomorphic(CyclotomicField(8))

False

"""

if is_CyclotomicField(other):

return self.zeta_order() == other.zeta_order()

return NumberField_generic.is_isomorphic(self, other)

def complex_embedding(self, prec=53):

r"""

Return the embedding of this cyclotomic field into the approximate

complex field with precision prec obtained by sending the generator

`\zeta` of self to exp(2\*pi\*i/n), where `n` is

the multiplicative order of `\zeta`.

EXAMPLES::

sage: C = CyclotomicField(4)

sage: C.complex_embedding()

Ring morphism:

From: Cyclotomic Field of order 4 and degree 2

To: Complex Field with 53 bits of precision

Defn: zeta4 |--> 6.12323399573677e-17 + 1.00000000000000*I

Note in the example above that the way zeta is computed (using sin

and cosine in MPFR) means that only the prec bits of the number

after the decimal point are valid.

sage: K = CyclotomicField(3)

sage: phi = K.complex_embedding(10)

sage: phi(K.0)

-0.50 + 0.87*I

sage: phi(K.0^3)

1.0

sage: phi(K.0^3 - 1)

0.00

sage: phi(K.0^3 + 7)

8.0

"""

CC = sage.rings.complex_field.ComplexField(prec)

return self.hom([CC.zeta(self._n())], check=False)

def complex_embeddings(self, prec=53):

r"""

Return all embeddings of this cyclotomic field into the approximate

complex field with precision prec.

If you want 53-bit double precision, which is faster but less

reliable, then do ``self.embeddings(CDF)``.

EXAMPLES::

sage: CyclotomicField(5).complex_embeddings()

[

Ring morphism:

From: Cyclotomic Field of order 5 and degree 4

To: Complex Field with 53 bits of precision

Defn: zeta5 |--> 0.309016994374947 + 0.951056516295154*I,

Ring morphism:

From: Cyclotomic Field of order 5 and degree 4

To: Complex Field with 53 bits of precision

Defn: zeta5 |--> -0.809016994374947 + 0.587785252292473*I,

Ring morphism:

From: Cyclotomic Field of order 5 and degree 4

To: Complex Field with 53 bits of precision

Defn: zeta5 |--> -0.809016994374947 - 0.587785252292473*I,

Ring morphism:

From: Cyclotomic Field of order 5 and degree 4

To: Complex Field with 53 bits of precision

Defn: zeta5 |--> 0.309016994374947 - 0.951056516295154*I

]

"""

CC = sage.rings.complex_field.ComplexField(prec)

try:

return self.__embeddings[CC]

except AttributeError:

self.__embeddings = {}

except KeyError:

pass

n = self._n()

z = CC.zeta(n)

X = [m for m in range(n) if arith.gcd(m,n) == 1]

v = [self.hom([z**n], check=False) for n in X]

self.__embeddings[CC] = Sequence(v, cr=True, immutable=True,

check=False, universe=self.Hom(CC))

return self.__embeddings[CC]

def real_embeddings(self, prec=53):

r"""

Return all embeddings of this cyclotomic field into the approximate

real field with precision prec.

Mostly, of course, there are no such embeddings.

EXAMPLES::

sage: CyclotomicField(4).real_embeddings()

[]

sage: CyclotomicField(2).real_embeddings()

[

Ring morphism:

From: Cyclotomic Field of order 2 and degree 1

To: Real Field with 53 bits of precision

Defn: -1 |--> -1.00000000000000

]

"""

K = sage.rings.real_mpfr.RealField(prec)

n = self._n()

if n > 2:

return Sequence([], cr=False, immutable=True,

check=False, universe=self.Hom(K))

else:

return self.embeddings(K)

def signature(self):

"""

Return (r1, r2), where r1 and r2 are the number of real embeddings

and pairs of complex embeddings of this cyclotomic field,

respectively.

Trivial since, apart from QQ, cyclotomic fields are totally

complex.

EXAMPLES::

sage: CyclotomicField(5).signature()

(0, 2)

sage: CyclotomicField(2).signature()

(1, 0)

"""

m = ZZ(self.degree())

if m == 1:

return (ZZ(1), ZZ(0))

else:

return (ZZ(0), ZZ(m/2))

def different(self):

"""

Returns the different ideal of the cyclotomic field self.

EXAMPLES::

sage: C20 = CyclotomicField(20)

sage: C20.different()

Fractional ideal (10, 2*zeta20^6 - 4*zeta20^4 - 4*zeta20^2 + 2)

sage: C18 = CyclotomicField(18)

sage: D = C18.different().norm()

sage: D == C18.discriminant().abs()

True

"""

try:

return self.__different

except AttributeError:

z = self.gen()

n = self._n()

D = self.ideal(1)

factors = n.factor()

for f in factors:

p = f[0]

r = f[1]

e = (r*p - r - 1)*p**(r-1)

D *= self.ideal(z**(n/p**r) - 1)**e

self.__different = D

return self.__different

def discriminant(self, v=None):

"""

Returns the discriminant of the ring of integers of the cyclotomic

field self, or if v is specified, the determinant of the trace

pairing on the elements of the list v.

Uses the formula for the discriminant of a prime power cyclotomic

field and Hilbert Theorem 88 on the discriminant of composita.

INPUT:

- ``v (optional)`` - list of element of this number

field

OUTPUT: Integer if v is omitted, and Rational otherwise.

EXAMPLES::

sage: CyclotomicField(20).discriminant()

4000000

sage: CyclotomicField(18).discriminant()

-19683

"""

if v is None:

try:

return self.__disc

except AttributeError:

n = self._n()

deg = self.degree()

d = ZZ(1) # so that CyclotomicField(1).disc() has the right type

factors = n.factor()

for (p, r) in factors:

e = (r*p - r - 1) * deg // (p-1)

d *= p**e

sign = 1

if len(factors) == 1 and (n == 4 or factors[0][0].mod(4) == 3):

sign = -1

elif len(factors) == 2 and factors[0] == (2, 1) and factors[1][0].mod(4) == 3:

sign = -1

self.__disc = sign*d

return self.__disc

else:

return NumberField_generic.discriminant(self, v)

def next_split_prime(self, p=2):

"""

Return the next prime integer `p` that splits completely in

this cyclotomic field (and does not ramify).

EXAMPLES::

sage: K.<z> = CyclotomicField(3)

sage: K.next_split_prime(7)

"""

n = self._n()

while True:

p = arith.next_prime(p)

if p % n == 1:

return p

def _pari_integral_basis(self, v=None, important=True):

"""

Internal function returning an integral basis of this number field in

PARI format.

This field is cyclotomic, so this is a trivial computation,

since the power basis on the generator is an integral basis.

Thus the ``v`` and ``important`` parameters are ignored.

EXAMPLES::

sage: CyclotomicField(5)._pari_integral_basis()

[1, y, y^2, y^3]

sage: len(CyclotomicField(137)._pari_integral_basis())

136

"""

try:

return self._integral_basis_dict[tuple()]

except KeyError:

z = pari(self.gen())

a = pari(1)

B = []

for n in range(self.degree()):

B.append(a.lift())

a *= z

self._integral_basis_dict[tuple()] = pari(B)

return B

def zeta_order(self):

"""

Return the order of the maximal root of unity contained in this

cyclotomic field.

EXAMPLES::

sage: CyclotomicField(1).zeta_order()

sage: CyclotomicField(4).zeta_order()

sage: CyclotomicField(5).zeta_order()

sage: CyclotomicField(5)._n()

sage: CyclotomicField(389).zeta_order()

778

"""

return self.__zeta_order

def _multiplicative_order_table(self):

"""

Return a dictionary that maps powers of zeta to their order. This

makes computing the orders of the elements of finite order in this

field faster.

EXAMPLES::

sage: v = CyclotomicField(6)._multiplicative_order_table()

sage: w = v.items(); w.sort(); w

[(-1, 2), (1, 1), (-x, 3), (-x + 1, 6), (x - 1, 3), (x, 6)]

"""

try:

return self.__multiplicative_order_table

except AttributeError:

t = {}

x = self(1)

n = self.zeta_order()

m = 0

zeta = self.zeta(n)

# todo: this desperately needs to be optimized!!!

for i in range(n):

t[x.polynomial()] = n//arith.GCD(m,n) # multiplicative_order of (zeta_n)**m

x *= zeta

m += 1

self.__multiplicative_order_table = t

return t

def zeta(self, n=None, all=False):

"""

Return an element of multiplicative order `n` in this

cyclotomic field.

If there is no such element, raise a ``ValueError``.

INPUT:

- ``n`` -- integer (default: None, returns element of

maximal order)

- ``all`` -- bool (default: False) - whether to return

a list of all primitive `n`-th roots of unity.

OUTPUT: root of unity or list

EXAMPLES::

sage: k = CyclotomicField(4)

sage: k.zeta()

zeta4

sage: k.zeta(2)

-1

sage: k.zeta().multiplicative_order()

sage: k = CyclotomicField(21)

sage: k.zeta().multiplicative_order()

sage: k.zeta(21).multiplicative_order()

sage: k.zeta(7).multiplicative_order()

sage: k.zeta(6).multiplicative_order()

sage: k.zeta(84)

Traceback (most recent call last):

ValueError: 84 does not divide order of generator (42)

sage: K.<a> = CyclotomicField(7)

sage: K.zeta(all=True)

[-a^4, -a^5, a^5 + a^4 + a^3 + a^2 + a + 1, -a, -a^2, -a^3]

sage: K.zeta(14, all=True)

[-a^4, -a^5, a^5 + a^4 + a^3 + a^2 + a + 1, -a, -a^2, -a^3]

sage: K.zeta(2, all=True)

[-1]

sage: K.<a> = CyclotomicField(10)

sage: K.zeta(20, all=True)

Traceback (most recent call last):

...

ValueError: 20 does not divide order of generator (10)

sage: K.<a> = CyclotomicField(5)

sage: K.zeta(4)

Traceback (most recent call last):

...

ValueError: 4 does not divide order of generator (10)

sage: v = K.zeta(5, all=True); v

[a, a^2, a^3, -a^3 - a^2 - a - 1]

sage: [b^5 for b in v]

[1, 1, 1, 1]

"""

if n is None:

n = self.zeta_order()

else:

n = Integer(n)

z = self.gen()

m = self._n()

if n % 2 == 0 and m % 2 == 1:

# In the n-th cyclotomic field, n odd, there are

# actually 2*n-th roots of unity, so we include them.

z = -z**((m+1)//2) # -z

m = 2*m

if m % n != 0:

raise ValueError("%s does not divide order of generator (%s)" %

(n, self.zeta_order()))

a = z**(m//n)

if not all:

return a

v = [a]

b = a*a

for i in range(2,n):

if n.gcd(i).is_one():

v.append(b)

b = b * a

return v

def number_of_roots_of_unity(self):

"""

Return number of roots of unity in this cyclotomic field.

EXAMPLES::

sage: K.<a> = CyclotomicField(21)

sage: K.number_of_roots_of_unity()

"""

n = self._n()

if n%2:

n *= 2

return n

def roots_of_unity(self):

"""

Return all the roots of unity in this cyclotomic field, primitive

or not.

EXAMPLES::

sage: K.<a> = CyclotomicField(3)

sage: zs = K.roots_of_unity(); zs

[1, a, -a - 1, -1, -a, a + 1]

sage: [ z**K.number_of_roots_of_unity() for z in zs ]

[1, 1, 1, 1, 1, 1]

"""

z = self.gen()

n = self._n()

v = [z**k for k in range(n)]

if n%2:

v += [-x for x in v]

return v

class NumberField_quadratic(NumberField_absolute):

r"""

Create a quadratic extension of the rational field.

The command ``QuadraticField(a)`` creates the field `\QQ(\sqrt{a})`.

EXAMPLES::

sage: QuadraticField(3, 'a')

Number Field in a with defining polynomial x^2 - 3

sage: QuadraticField(-4, 'b')

Number Field in b with defining polynomial x^2 + 4

"""

def __init__(self, polynomial, name=None, latex_name=None, check=True, embedding=None,

assume_disc_small=False, maximize_at_primes=None, structure=None):

"""

Create a quadratic number field.

EXAMPLES::

sage: k.<a> = QuadraticField(5, check=False); k

Number Field in a with defining polynomial x^2 - 5

Don't do this::

sage: k.<a> = QuadraticField(4, check=False); k

Number Field in a with defining polynomial x^2 - 4

TESTS::

sage: k.<a> = QuadraticField(7)

sage: type(k.zero())

sage: type(k.one())

sage: TestSuite(k).run()

Check that :trac:`23008` is fixed::

sage: z = polygen(ZZ, 'z')

sage: K.<phi> = NumberField(z^2 - z - 1, embedding=QQbar(golden_ratio))

sage: floor(phi)

"""

NumberField_absolute.__init__(self, polynomial, name=name, check=check,

embedding=embedding, latex_name=latex_name,

assume_disc_small=assume_disc_small, maximize_at_primes=maximize_at_primes, structure=structure)

self._standard_embedding = True

self._element_class = number_field_element_quadratic.NumberFieldElement_quadratic

c, b, a = [QQ(t) for t in self.defining_polynomial().list()]

# set the generator

Dpoly = b*b - 4*a*c

D = (Dpoly.numer() * Dpoly.denom()).squarefree_part(bound=10000)

self._D = D

parts = -b/(2*a), (Dpoly/D).sqrt()/(2*a)

self._NumberField_generic__gen = self._element_class(self, parts)

# we must set the flag _standard_embedding *before* any element creation

# Note that in the following code, no element is built.

if self.coerce_embedding() is not None and CDF.has_coerce_map_from(self):

rootD = CDF(number_field_element_quadratic.NumberFieldElement_quadratic(self, (QQ(0),QQ(1))))

if D > 0:

self._standard_embedding = rootD.real() > 0

else:

self._standard_embedding = rootD.imag() > 0

# we reset _NumberField_generic__gen has the flag standard_embedding

# might be modified

self._NumberField_generic__gen = self._element_class(self, parts)

# NumberField_absolute.__init__(...) set _zero_element and

# _one_element to NumberFieldElement_absolute values, which is

# wrong (and dangerous; such elements can actually be used to

# crash Sage: see #5316). Overwrite them with correct values.

self._zero_element = self._element_class(self, (QQ(0), QQ(0)))

self._one_element = self._element_class(self, (QQ(1), QQ(0)))

def _coerce_map_from_(self, K):

"""

EXAMPLES::

sage: K.<a> = QuadraticField(-3)

sage: f = K.coerce_map_from(QQ); f # indirect doctest

Natural morphism:

From: Rational Field

To: Number Field in a with defining polynomial x^2 + 3

sage: f(3/5)

3/5

sage: parent(f(3/5)) is K

True

sage: g = K.coerce_map_from(ZZ); g # indirect doctest

Natural morphism:

From: Integer Ring

To: Number Field in a with defining polynomial x^2 + 3

sage: g(1)

sage: parent(g(1)) is K

True

"""

if K is ZZ:

return number_field_element_quadratic.Z_to_quadratic_field_element(self)

if K is int:

return self._coerce_map_via([ZZ], int) # faster than direct

if K is QQ:

return number_field_element_quadratic.Q_to_quadratic_field_element(self)

return NumberField_absolute._coerce_map_from_(self, K)

def _latex_(self):

"""

Return the latex representation of this quadratic field.

EXAMPLES::

sage: Z = QuadraticField(7)

sage: latex(Z) # indirect doctest

\Bold{Q}(\sqrt{7})

sage: Z = QuadraticField(7, latex_name='x')

sage: latex(Z) # indirect doctest

\Bold{Q}[x]/(x^{2} - 7)

"""

v = self.latex_variable_name()

if v.startswith('\\sqrt'):

return "%s(%s)"%(latex(QQ), v)

else:

return NumberField_generic._latex_(self)

def _polymake_init_(self):

r"""

Return the polymake representation of this quadratic field.

This is merely a string, and does not represent a specific quadratic field.

In polymake, only the elements know which field they belong to.

EXAMPLES::

sage: Z = QuadraticField(7)

sage: polymake(Z) # optional - polymake # indirect doctest

QuadraticExtension

"""

return '"QuadraticExtension"'

def discriminant(self, v=None):

"""

Returns the discriminant of the ring of integers of the number

field, or if v is specified, the determinant of the trace pairing

on the elements of the list v.

INPUT:

- ``v (optional)`` - list of element of this number

field

OUTPUT: Integer if v is omitted, and Rational otherwise.

EXAMPLES::

sage: K. = NumberField(x^2+1)

sage: K.discriminant()

-4

sage: K.<a> = NumberField(x^2+5)

sage: K.discriminant()

-20

sage: K.<a> = NumberField(x^2-5)

sage: K.discriminant()

"""

if v is None:

try:

return self.__disc

except AttributeError:

d = self._D.squarefree_part()

if d % 4 != 1:

d *= 4

self.__disc = d

return self.__disc

else:

return NumberField_generic.discriminant(self, v)

def is_galois(self):

"""

Return True since all quadratic fields are automatically Galois.

EXAMPLES::

sage: QuadraticField(1234,'d').is_galois()

True

"""

return True

def class_number(self, proof=None):

r"""

Return the size of the class group of self.

If proof = False (*not* the default!) and the discriminant of the

field is negative, then the following warning from the PARI manual

applies:

.. warning::

For `D<0`, this function may give incorrect results when

the class group has a low exponent (has many cyclic

factors), because implementing Shank's method in full

generality slows it down immensely.

EXAMPLES::

sage: QuadraticField(-23,'a').class_number()

These are all the primes so that the class number of

`\QQ(\sqrt{-p})` is `1`::

sage: [d for d in prime_range(2,300) if not is_square(d) and QuadraticField(-d,'a').class_number() == 1]

[2, 3, 7, 11, 19, 43, 67, 163]

It is an open problem to *prove* that there are infinity many

positive square-free `d` such that

`\QQ(\sqrt{d})` has class number `1`:

sage: len([d for d in range(2,200) if not is_square(d) and QuadraticField(d,'a').class_number() == 1])

121

TESTS::

sage: type(QuadraticField(-23,'a').class_number())

sage: type(NumberField(x^3 + 23, 'a').class_number())

sage: type(NumberField(x^3 + 23, 'a').extension(x^2 + 5, 'b').class_number())

sage: type(CyclotomicField(10).class_number())

"""

proof = proof_flag(proof)

try:

return self.__class_number

except AttributeError:

self.__class_number = self.discriminant().class_number(proof)

return self.__class_number

def hilbert_class_field_defining_polynomial(self, name='x'):

r"""

Returns a polynomial over `\QQ` whose roots generate the

Hilbert class field of this quadratic field as an extension of

this quadratic field.

.. note::

Computed using PARI via Schertz's method. This

implementation is quite fast.

EXAMPLES::

sage: K. = QuadraticField(-23)

sage: K.hilbert_class_field_defining_polynomial()

x^3 - x^2 + 1

Note that this polynomial is not the actual Hilbert class

polynomial: see ``hilbert_class_polynomial``::

sage: K.hilbert_class_polynomial()

x^3 + 3491750*x^2 - 5151296875*x + 12771880859375

sage: K.<a> = QuadraticField(-431)

sage: K.class_number()

sage: K.hilbert_class_field_defining_polynomial(name='z')

z^21 + 6*z^20 + 9*z^19 - 4*z^18 + 33*z^17 + 140*z^16 + 220*z^15 + 243*z^14 + 297*z^13 + 461*z^12 + 658*z^11 + 743*z^10 + 722*z^9 + 681*z^8 + 619*z^7 + 522*z^6 + 405*z^5 + 261*z^4 + 119*z^3 + 35*z^2 + 7*z + 1

"""

f = pari(self.discriminant()).quadhilbert()

return QQ[name](f)

def hilbert_class_field(self, names):

r"""

Returns the Hilbert class field of this quadratic field as a

relative extension of this field.

.. note::

For the polynomial that defines this field as a relative

extension, see the ``hilbert_class_field_defining_polynomial``

command, which is vastly faster than this command, since it doesn't

construct a relative extension.

EXAMPLES::

sage: K.<a> = NumberField(x^2 + 23)

sage: L = K.hilbert_class_field('b'); L

Number Field in b with defining polynomial x^3 - x^2 + 1 over its base field

sage: L.absolute_field('c')

Number Field in c with defining polynomial x^6 - 2*x^5 + 70*x^4 - 90*x^3 + 1631*x^2 - 1196*x + 12743

sage: K.hilbert_class_field_defining_polynomial()

x^3 - x^2 + 1

"""

f = self.hilbert_class_field_defining_polynomial()

return self.extension(f, names)

def hilbert_class_polynomial(self, name='x'):

r"""

Compute the Hilbert class polynomial of this quadratic field.

Right now, this is only implemented for imaginary quadratic

fields.

EXAMPLES::

sage: K.<a> = QuadraticField(-3)

sage: K.hilbert_class_polynomial()

sage: K.<a> = QuadraticField(-31)

sage: K.hilbert_class_polynomial(name='z')

z^3 + 39491307*z^2 - 58682638134*z + 1566028350940383

"""

D = self.discriminant()

if D > 0:

raise NotImplementedError("Hilbert class polynomial is not implemented for real quadratic fields.")

from sage.schemes.elliptic_curves.all import hilbert_class_polynomial as HCP

return QQ[name](HCP(D))

def is_fundamental_discriminant(D):

r"""

Return True if the integer `D` is a fundamental

discriminant, i.e., if `D \cong 0,1\pmod{4}`, and

`D\neq 0, 1` and either (1) `D` is square free or

(2) we have `D\cong 0\pmod{4}` with

`D/4 \cong 2,3\pmod{4}` and `D/4` square free. These

are exactly the discriminants of quadratic fields.

EXAMPLES::

sage: [D for D in range(-15,15) if is_fundamental_discriminant(D)]

[-15, -11, -8, -7, -4, -3, 5, 8, 12, 13]

sage: [D for D in range(-15,15) if not is_square(D) and QuadraticField(D,'a').disc() == D]

[-15, -11, -8, -7, -4, -3, 5, 8, 12, 13]

"""

d = D % 4

if not (d in [0,1]):

return False

return D != 1 and D != 0 and \

(arith.is_squarefree(D) or \

(d == 0 and (D//4)%4 in [2,3] and arith.is_squarefree(D//4)))

###################

# For pickling

###################

def NumberField_absolute_v1(poly, name, latex_name, canonical_embedding=None):

"""

Used for unpickling old pickles.

EXAMPLES::

sage: from sage.rings.number_field.number_field import NumberField_absolute_v1

sage: R.<x> = QQ[]

sage: NumberField_absolute_v1(x^2 + 1, 'i', 'i')

Number Field in i with defining polynomial x^2 + 1

"""

return NumberField(polynomial=poly, name=name, latex_name=latex_name, check=False, embedding=canonical_embedding)

NumberField_generic_v1 = NumberField_absolute_v1 # for historical reasons only (so old objects unpickle)

def NumberField_cyclotomic_v1(zeta_order, name, canonical_embedding=None):

"""

Used for unpickling old pickles.

EXAMPLES::

sage: from sage.rings.number_field.number_field import NumberField_cyclotomic_v1

sage: NumberField_cyclotomic_v1(5,'a')

Cyclotomic Field of order 5 and degree 4

sage: NumberField_cyclotomic_v1(5,'a').variable_name()

'a'

"""

return CyclotomicField(n=zeta_order, names=name, embedding=canonical_embedding)

def NumberField_quadratic_v1(poly, name, canonical_embedding=None):

"""

Used for unpickling old pickles.

EXAMPLES::

sage: from sage.rings.number_field.number_field import NumberField_quadratic_v1

sage: R.<x> = QQ[]

sage: NumberField_quadratic_v1(x^2 - 2, 'd')

Number Field in d with defining polynomial x^2 - 2

"""

return NumberField(polynomial=poly, name=name, check=False, embedding=canonical_embedding)

def put_natural_embedding_first(v):

"""

Helper function for embeddings() functions for number fields.

INPUT: a list of embeddings of a number field

OUTPUT: None. The

list is altered in-place, so that, if possible, the first embedding

has been switched with one of the others, so that if there is an

embedding which preserves the generator names then it appears

first.

EXAMPLES::

sage: K.<a> = CyclotomicField(7)

sage: embs = K.embeddings(K)

sage: [e(a) for e in embs] # random - there is no natural sort order

[a, a^2, a^3, a^4, a^5, -a^5 - a^4 - a^3 - a^2 - a - 1]

sage: id = [ e for e in embs if e(a) == a ][0]; id

Ring endomorphism of Cyclotomic Field of order 7 and degree 6

Defn: a |--> a

sage: permuted_embs = list(embs); permuted_embs.remove(id); permuted_embs.append(id)

sage: [e(a) for e in permuted_embs] # random - but natural map is not first

[a^2, a^3, a^4, a^5, -a^5 - a^4 - a^3 - a^2 - a - 1, a]

sage: permuted_embs[0] != a

True

sage: from sage.rings.number_field.number_field import put_natural_embedding_first

sage: put_natural_embedding_first(permuted_embs)

sage: [e(a) for e in permuted_embs] # random - but natural map is first

[a, a^3, a^4, a^5, -a^5 - a^4 - a^3 - a^2 - a - 1, a^2]

sage: permuted_embs[0] == id

True

"""

for i in range(len(v)):

phi = v[i]

a = str(list(phi.domain().gens()))

b = str(list(phi.im_gens()))

if a == b:

v[i] = v[0]

v[0] = phi

return

def refine_embedding(e, prec=None):

r"""

Given an embedding from a number field to either `\RR` or

`\CC`, returns an equivalent embedding with higher precision.

INPUT:

- ``e`` - an embedding of a number field into either

RR or CC (with some precision)

- ``prec`` - (default None) the desired precision; if None,

current precision is doubled; if Infinity, the equivalent

embedding into either ``QQbar`` or ``AA`` is returned.

EXAMPLES::

sage: from sage.rings.number_field.number_field import refine_embedding

sage: K = CyclotomicField(3)

sage: e10 = K.complex_embedding(10)

sage: e10.codomain().precision()

sage: e25 = refine_embedding(e10, prec=25)

sage: e25.codomain().precision()

An example where we extend a real embedding into ``AA``::

sage: K.<a> = NumberField(x^3-2)

sage: K.signature()

(1, 1)

sage: e = K.embeddings(RR)[0]; e

Ring morphism:

From: Number Field in a with defining polynomial x^3 - 2

To: Real Field with 53 bits of precision

Defn: a |--> 1.25992104989487

sage: e = refine_embedding(e,Infinity); e

Ring morphism:

From: Number Field in a with defining polynomial x^3 - 2

To: Algebraic Real Field

Defn: a |--> 1.259921049894873?

Now we can obtain arbitrary precision values with no trouble::

sage: RealField(150)(e(a))

1.2599210498948731647672106072782283505702515

sage: _^3

2.0000000000000000000000000000000000000000000

sage: RealField(200)(e(a^2-3*a+7))

4.8076379022835799804500738174376232086807389337953290695624

Complex embeddings can be extended into ``QQbar``::

sage: e = K.embeddings(CC)[0]; e

Ring morphism:

From: Number Field in a with defining polynomial x^3 - 2

To: Complex Field with 53 bits of precision

Defn: a |--> -0.62996052494743... - 1.09112363597172*I

sage: e = refine_embedding(e,Infinity); e

Ring morphism:

From: Number Field in a with defining polynomial x^3 - 2

To: Algebraic Field

Defn: a |--> -0.6299605249474365? - 1.091123635971722?*I

sage: ComplexField(200)(e(a))

-0.62996052494743658238360530363911417528512573235075399004099 - 1.0911236359717214035600726141898088813258733387403009407036*I

sage: e(a)^3

Embeddings into lazy fields work::

sage: L = CyclotomicField(7)

sage: x = L.specified_complex_embedding(); x

Generic morphism:

From: Cyclotomic Field of order 7 and degree 6

To: Complex Lazy Field

Defn: zeta7 -> 0.623489801858734? + 0.781831482468030?*I

sage: refine_embedding(x, 300)

Ring morphism:

From: Cyclotomic Field of order 7 and degree 6

To: Complex Field with 300 bits of precision

Defn: zeta7 |--> 0.623489801858733530525004884004239810632274730896402105365549439096853652456487284575942507 + 0.781831482468029808708444526674057750232334518708687528980634958045091731633936441700868007*I

sage: refine_embedding(x, infinity)

Ring morphism:

From: Cyclotomic Field of order 7 and degree 6

To: Algebraic Field

Defn: zeta7 |--> 0.6234898018587335? + 0.7818314824680299?*I

When the old embedding is into the real lazy field,

then only real embeddings should be considered.

See :trac:`17495`::

sage: R.<x> = QQ[]

sage: K.<a> = NumberField(x^3 + x - 1, embedding=0.68)

sage: from sage.rings.number_field.number_field import refine_embedding

sage: refine_embedding(K.specified_complex_embedding(), 100)

Ring morphism:

From: Number Field in a with defining polynomial x^3 + x - 1

To: Real Field with 100 bits of precision

Defn: a |--> 0.68232780382801932736948373971

sage: refine_embedding(K.specified_complex_embedding(), Infinity)

Ring morphism:

From: Number Field in a with defining polynomial x^3 + x - 1

To: Algebraic Real Field

Defn: a |--> 0.6823278038280193?

"""

K = e.domain()

RC = e.codomain()

if RC in (sage.rings.qqbar.AA, sage.rings.qqbar.QQbar):

return e

if RC in (RLF, CLF):

prec_old = e.gen_image().approx().prec()

old_root = e(K.gen()).approx()

else:

prec_old = RC.precision()

old_root = e(K.gen())

if prec is None:

prec = 2*prec_old

elif prec_old >= prec:

return e

# We first compute all the embeddings at the new precision:

if sage.rings.real_mpfr.is_RealField(RC) or RC in (RDF, RLF):

if prec == Infinity:

elist = K.embeddings(sage.rings.qqbar.AA)

else:

elist = K.real_embeddings(prec)

else:

if prec == Infinity:

elist = K.embeddings(sage.rings.qqbar.QQbar)

else:

elist = K.complex_embeddings(prec)

# Now we determine which is an extension of the old one; this

# relies on the fact that coercing a high-precision root into a

# field with lower precision will equal the lower-precision root!

diffs = [(RC(ee(K.gen()))-old_root).abs() for ee in elist]

return elist[min(zip(diffs, count()))[1]]

def is_real_place(v):

r"""

Return ``True`` if ``v`` is real, ``False`` if ``v`` is complex

INPUT:

- ``v`` -- an infinite place of ``K``

OUTPUT:

A boolean indicating whether a place is real (``True``) or complex (``False``).

EXAMPLES::

sage: K.<xi> = NumberField(x^3-3)

sage: phi_real = K.places()[0]

sage: phi_complex = K.places()[1]

sage: v_fin = tuple(K.primes_above(3))[0]

sage: is_real_place(phi_real)

True

sage: is_real_place(phi_complex)

False

It is an error to put in a finite place

sage: is_real_place(v_fin)

Traceback (most recent call last):

...

AttributeError: 'NumberFieldFractionalIdeal' object has no attribute 'im_gens'

"""

RR = sage.rings.real_mpfr.RealField(53)

try:

RR(v.im_gens()[0])

return True

except TypeError:

return False

Coverage for local/lib/python2.7/site-packages/sage/rings/number_field/number_field.py : 81%

2193 statements 1776 run 417 missing 0 excluded