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r"""

Structure maps for number fields

Provides isomorphisms between relative and absolute presentations, to and from

vector spaces, name changing maps, etc.

EXAMPLES::

sage: L.<cuberoot2, zeta3> = CyclotomicField(3).extension(x^3 - 2)

sage: K = L.absolute_field('a')

sage: from_K, to_K = K.structure()

sage: from_K

Isomorphism map:

From: Number Field in a with defining polynomial x^6 - 3*x^5 + 6*x^4 - 11*x^3 + 12*x^2 + 3*x + 1

To: Number Field in cuberoot2 with defining polynomial x^3 - 2 over its base field

sage: to_K

Isomorphism map:

From: Number Field in cuberoot2 with defining polynomial x^3 - 2 over its base field

To: Number Field in a with defining polynomial x^6 - 3*x^5 + 6*x^4 - 11*x^3 + 12*x^2 + 3*x + 1

"""

#*****************************************************************************

# Distributed under the terms of the GNU General Public License (GPL)

# This code is distributed in the hope that it will be useful,

# but WITHOUT ANY WARRANTY; without even the implied warranty of

# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU

# General Public License for more details.

# The full text of the GPL is available at:

# http://www.gnu.org/licenses/

#*****************************************************************************

from sage.categories.map import Map

from sage.categories.homset import Hom

from sage.categories.morphism import IdentityMorphism

import sage.rings.rational_field as rational_field

from sage.libs.pari.all import pari

QQ = rational_field.RationalField()

IdentityMap = IdentityMorphism

class NumberFieldIsomorphism(Map):

r"""

A base class for various isomorphisms between number fields and

vector spaces.

EXAMPLES::

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

sage: V, fr, to = K.vector_space()

sage: isinstance(fr, sage.rings.number_field.maps.NumberFieldIsomorphism)

True

"""

def _repr_type(self):

r"""

EXAMPLES::

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

sage: V, fr, to = K.vector_space()

sage: fr._repr_type()

'Isomorphism'

"""

return "Isomorphism"

def is_injective(self):

r"""

EXAMPLES::

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

sage: V, fr, to = K.vector_space()

sage: fr.is_injective()

True

"""

return True

def is_surjective(self):

r"""

EXAMPLES::

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

sage: V, fr, to = K.vector_space()

sage: fr.is_surjective()

True

"""

return True

class MapVectorSpaceToNumberField(NumberFieldIsomorphism):

r"""

The map to an absolute number field from its underlying `\QQ`-vector space.

EXAMPLES::

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

sage: V, fr, to = K.vector_space()

sage: V

Vector space of dimension 4 over Rational Field

sage: fr

Isomorphism map:

From: Vector space of dimension 4 over Rational Field

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

sage: to

Isomorphism map:

From: Number Field in a with defining polynomial x^4 + 3*x + 1

To: Vector space of dimension 4 over Rational Field

sage: type(fr), type(to)

(<class 'sage.rings.number_field.maps.MapVectorSpaceToNumberField'>,

<class 'sage.rings.number_field.maps.MapNumberFieldToVectorSpace'>)

sage: fr.is_injective(), fr.is_surjective()

(True, True)

sage: fr.domain(), to.codomain()

(Vector space of dimension 4 over Rational Field, Vector space of dimension 4 over Rational Field)

sage: to.domain(), fr.codomain()

(Number Field in a with defining polynomial x^4 + 3*x + 1, Number Field in a with defining polynomial x^4 + 3*x + 1)

sage: fr * to

Composite map:

From: Number Field in a with defining polynomial x^4 + 3*x + 1

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

Defn: Isomorphism map:

From: Number Field in a with defining polynomial x^4 + 3*x + 1

To: Vector space of dimension 4 over Rational Field

then

Isomorphism map:

From: Vector space of dimension 4 over Rational Field

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

sage: to * fr

Composite map:

From: Vector space of dimension 4 over Rational Field

To: Vector space of dimension 4 over Rational Field

Defn: Isomorphism map:

From: Vector space of dimension 4 over Rational Field

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

then

Isomorphism map:

From: Number Field in a with defining polynomial x^4 + 3*x + 1

To: Vector space of dimension 4 over Rational Field

sage: to(a), to(a + 1)

((0, 1, 0, 0), (1, 1, 0, 0))

sage: fr(to(a)), fr(V([0, 1, 2, 3]))

(a, 3*a^3 + 2*a^2 + a)

"""

def __init__(self, V, K):

r"""

EXAMPLES::

sage: K.<c> = NumberField(x^9 + 3)

sage: V, fr, to = K.vector_space(); fr # indirect doctest

Isomorphism map:

From: Vector space of dimension 9 over Rational Field

To: Number Field in c with defining polynomial x^9 + 3

sage: type(fr)

"""

NumberFieldIsomorphism.__init__(self, Hom(V, K))

def _call_(self, v):

r"""

EXAMPLES::

sage: K.<c> = NumberField(x^9 + 3)

sage: V, fr, to = K.vector_space()

sage: list(map(fr, V.gens())) # indirect doctest

[1, c, c^2, c^3, c^4, c^5, c^6, c^7, c^8]

"""

K = self.codomain()

f = K.polynomial_ring()(v.list())

return K._element_class(K, f)

class MapNumberFieldToVectorSpace(Map):

r"""

A class for the isomorphism from an absolute number field to its underlying

`\QQ`-vector space.

EXAMPLES::

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

sage: V, fr, to = L.vector_space()

sage: type(to)

"""

def __init__(self, K, V):

r"""

Standard initialisation function.

EXAMPLES::

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

sage: L.vector_space()[2] # indirect doctest

Isomorphism map:

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

To: Vector space of dimension 3 over Rational Field

"""

NumberFieldIsomorphism.__init__(self, Hom(K, V))

def _repr_type(self):

r"""

EXAMPLES::

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

sage: V, fr, to = L.relative_vector_space()

sage: fr._repr_type()

'Isomorphism'

"""

return "Isomorphism"

def _call_(self, x):

r"""

EXAMPLES::

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

sage: V, _, to = L.vector_space()

sage: v = to(a^2 - a/37 + 56); v # indirect doctest

(56, -1/37, 1)

sage: v.parent() is V

True

"""

v = x._coefficients()

k = self.domain().degree() - len(v)

if k > 0:

v = v + [QQ.zero()] * k

return self.codomain()(v)

class MapRelativeVectorSpaceToRelativeNumberField(NumberFieldIsomorphism):

r"""

EXAMPLES::

sage: L. = NumberField(x^4 + 3*x^2 + 1)

sage: K = L.relativize(L.subfields(2)[0][1], 'a'); K

Number Field in a with defining polynomial x^2 - b0*x + 1 over its base field

sage: V, fr, to = K.relative_vector_space()

sage: V

Vector space of dimension 2 over Number Field in b0 with defining polynomial x^2 + 1

sage: fr

Isomorphism map:

From: Vector space of dimension 2 over Number Field in b0 with defining polynomial x^2 + 1

To: Number Field in a with defining polynomial x^2 - b0*x + 1 over its base field

sage: type(fr)

sage: a0 = K.gen(); b0 = K.base_field().gen()

sage: fr(to(a0 + 2*b0)), fr(V([0, 1])), fr(V([b0, 2*b0]))

(a + 2*b0, a, 2*b0*a + b0)

sage: (fr * to)(K.gen()) == K.gen()

True

sage: (to * fr)(V([1, 2])) == V([1, 2])

True

"""

def __init__(self, V, K):

r"""

EXAMPLES::

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

sage: V, _, to = K.relative_vector_space(); to # indirect doctest

Isomorphism map:

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

To: Vector space of dimension 2 over Number Field in b with defining polynomial x^2 - 2

"""

NumberFieldIsomorphism.__init__(self, Hom(V, K))

def _call_(self, v):

r"""

EXAMPLES::

sage: L. = NumberField(x^4 + 3*x^2 + 1)

sage: K = L.relativize(L.subfields(2)[0][1], 'a')

sage: a0 = K.gen(); b0 = K.base_field().gen()

sage: V, fr, to = K.relative_vector_space()

sage: fr(to(a0 + 2*b0)), fr(V([0, 1])), fr(V([b0, 2*b0])) # indirect doctest

(a + 2*b0, a, 2*b0*a + b0)

"""

K = self.codomain()

B = K.base_field().absolute_field('a')

# Convert v to a PARI polynomial in x with coefficients that

# are polynomials in y.

_, to_B = B.structure()

h = pari([to_B(a).__pari__('y') for a in v]).Polrev()

# Rewrite the polynomial in terms of an absolute generator for

# the relative number field.

g = K._pari_rnfeq()._eltreltoabs(h)

return K._element_class(K, g)

class MapRelativeNumberFieldToRelativeVectorSpace(NumberFieldIsomorphism):

r"""

EXAMPLES::

sage: K.<a, b> = NumberField([x^3 - x + 1, x^2 + 23])

sage: V, fr, to = K.relative_vector_space()

sage: type(to)

"""

def __init__(self, K, V):

r"""

EXAMPLES::

sage: L. = NumberField(x^4 + 3*x^2 + 1)

sage: K = L.relativize(L.subfields(2)[0][1], 'a')

sage: V, fr, to = K.relative_vector_space()

sage: to

Isomorphism map:

From: Number Field in a with defining polynomial x^2 - b0*x + 1 over its base field

To: Vector space of dimension 2 over Number Field in b0 with defining polynomial x^2 + 1

"""

NumberFieldIsomorphism.__init__(self, Hom(K, V))

def _call_(self, alpha):

"""

TESTS::

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

sage: R.<y> = K[]

sage: D.<x0> = K.extension(y + a + 1)

sage: D(a)

sage: V, from_V, to_V = D.relative_vector_space()

sage: to_V(a) # indirect doctest

(a)

sage: to_V(a^3) # indirect doctest

(a^3)

sage: to_V(x0) # indirect doctest

(-a - 1)

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

sage: L. = K.extension(x-5)

sage: L(a)

sage: a*b

5*a

sage: b

sage: V, from_V, to_V = L.relative_vector_space()

sage: to_V(a) # indirect doctest

(a)

"""

K = self.domain()

# The element alpha is represented internally by an absolute

# polynomial over QQ, and f is its PARI representation.

f = alpha._pari_polynomial('x')

# Convert f to a relative polynomial g; this is a polynomial

# in x whose coefficients are polynomials in y.

g = K._pari_rnfeq()._eltabstorel_lift(f)

# Now g is a polynomial in the standard generator of the PARI

# field; convert it to a polynomial in the Sage generator.

if g.poldegree() > 0:

beta = K._pari_relative_structure()[2]

g = g(beta).lift()

# Convert the coefficients to elements of the base field.

B, from_B, _ = K.absolute_base_field()

return self.codomain()([from_B(B(z.lift(), check=False)) for z in g.Vecrev(-K.relative_degree())])

class NameChangeMap(NumberFieldIsomorphism):

r"""

A map between two isomorphic number fields with the same defining

polynomial but different variable names.

EXAMPLES::

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

sage: L. = K.change_names()

sage: from_L, to_L = L.structure()

sage: from_L

Isomorphism given by variable name change map:

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

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

sage: to_L

Isomorphism given by variable name change map:

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

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

sage: type(from_L), type(to_L)

(<class 'sage.rings.number_field.maps.NameChangeMap'>, <class 'sage.rings.number_field.maps.NameChangeMap'>)

"""

def __init__(self, K, L):

r"""

EXAMPLES::

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

sage: L.<c, d> = K.change_names()

sage: L.structure()

(Isomorphism given by variable name change map:

From: Number Field in c with defining polynomial x^2 - 3 over its base field

To: Number Field in a with defining polynomial x^2 - 3 over its base field, Isomorphism given by variable name change map:

From: Number Field in a with defining polynomial x^2 - 3 over its base field

To: Number Field in c with defining polynomial x^2 - 3 over its base field)

"""

NumberFieldIsomorphism.__init__(self, Hom(K, L))

def _repr_type(self):

r"""

EXAMPLES::

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

sage: L. = K.change_names()

sage: from_L, to_L = L.structure()

sage: from_L._repr_type()

'Isomorphism given by variable name change'

"""

return "Isomorphism given by variable name change"

def _call_(self, x):

r"""

EXAMPLES::

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

sage: L.<c, d> = K.change_names()

sage: to_K, from_K = L.structure()

sage: from_K(a + 17*b) # indirect doctest

c + 17*d

sage: to_K(57*c + 19/8*d) # indirect doctest

57*a + 19/8*b

"""

y = x.__copy__()

y._set_parent(self.codomain())

return y

class MapRelativeToAbsoluteNumberField(NumberFieldIsomorphism):

r"""

EXAMPLES::

sage: K.<a> = NumberField(x^6 + 4*x^2 + 200)

sage: L = K.relativize(K.subfields(3)[0][1], 'b'); L

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

sage: fr, to = L.structure()

sage: fr

Relative number field morphism:

From: Number Field in b with defining polynomial x^2 + a0 over its base field

To: Number Field in a with defining polynomial x^6 + 4*x^2 + 200

Defn: b |--> a

a0 |--> -a^2

sage: to

Ring morphism:

From: Number Field in a with defining polynomial x^6 + 4*x^2 + 200

To: Number Field in b with defining polynomial x^2 + a0 over its base field

Defn: a |--> b

sage: type(fr), type(to)

(<class 'sage.rings.number_field.morphism.RelativeNumberFieldHomomorphism_from_abs'>,

<class 'sage.rings.number_field.morphism.NumberFieldHomomorphism_im_gens'>)

sage: M.<c> = L.absolute_field(); M

Number Field in c with defining polynomial x^6 + 4*x^2 + 200

sage: fr, to = M.structure()

sage: fr

Isomorphism map:

From: Number Field in c with defining polynomial x^6 + 4*x^2 + 200

To: Number Field in b with defining polynomial x^2 + a0 over its base field

sage: to

Isomorphism map:

From: Number Field in b with defining polynomial x^2 + a0 over its base field

To: Number Field in c with defining polynomial x^6 + 4*x^2 + 200

sage: type(fr), type(to)

(<class 'sage.rings.number_field.maps.MapAbsoluteToRelativeNumberField'>,

<class 'sage.rings.number_field.maps.MapRelativeToAbsoluteNumberField'>)

sage: fr(M.gen()), to(fr(M.gen())) == M.gen()

(b, True)

sage: to(L.gen()), fr(to(L.gen())) == L.gen()

(c, True)

sage: (to * fr)(M.gen()) == M.gen(), (fr * to)(L.gen()) == L.gen()

(True, True)

"""

def __init__(self, R, A):

r"""

EXAMPLES::

sage: L.<a, b> = NumberField([x^2 + 3, x^2 + 5])

sage: K.<c> = L.absolute_field()

sage: f = K.structure()[1]; f

Isomorphism map:

From: Number Field in a with defining polynomial x^2 + 3 over its base field

To: Number Field in c with defining polynomial x^4 + 16*x^2 + 4

sage: type(f)

"""

NumberFieldIsomorphism.__init__(self, Hom(R, A))

def _call_(self, x):

r"""

EXAMPLES::

sage: L.<a, b> = NumberField([x^2 + 3, x^2 + 5])

sage: K.<c> = L.absolute_field()

sage: f = K.structure()[1]

sage: f(a + 3*b) # indirect doctest

-c^3 - 17*c

"""

A = self.codomain() # absolute field

f = x.polynomial()

return A._element_class(A, f)

class MapAbsoluteToRelativeNumberField(NumberFieldIsomorphism):

r"""

See :class:`~MapRelativeToAbsoluteNumberField` for examples.

"""

def __init__(self, A, R):

r"""

EXAMPLES::

sage: L.<a, b> = NumberField([x^2 + 3, x^2 + 5])

sage: K.<c> = L.absolute_field()

sage: f = K.structure()[0] # indirect doctest

sage: type(f)

"""

NumberFieldIsomorphism.__init__(self, Hom(A, R))

def _call_(self, x):

r"""

EXAMPLES::

sage: L.<a, b> = NumberField([x^2 + 3, x^2 + 5])

sage: K.<c> = L.absolute_field()

sage: f = K.structure()[0]

sage: f(c + 13*c^2) # indirect doctest

(-26*b + 1)*a - b - 104

"""

R = self.codomain() # relative field

f = x.polynomial()

return R._element_class(R, f)

class MapVectorSpaceToRelativeNumberField(NumberFieldIsomorphism):

r"""

The isomorphism to a relative number field from its underlying `\QQ`-vector

space. Compare :class:`~MapRelativeVectorSpaceToRelativeNumberField`.

EXAMPLES::

sage: L.<a, b> = NumberField([x^2 + 3, x^2 + 5])

sage: V, fr, to = L.absolute_vector_space()

sage: type(fr)

"""

def __init__(self, V, L, from_V, from_K):

r"""

EXAMPLES::

sage: L.<a, b> = NumberField([x^2 + 3, x^2 + 5])

sage: V, fr, to = L.absolute_vector_space() # indirect doctest

sage: fr

Isomorphism map:

From: Vector space of dimension 4 over Rational Field

To: Number Field in a with defining polynomial x^2 + 3 over its base field

"""

self.__from_V = from_V

self.__from_K = from_K

NumberFieldIsomorphism.__init__(self, Hom(V, L))

def _call_(self, x):

r"""

EXAMPLES::

sage: L.<a, b> = NumberField([x^2 + 3, x^2 + 5])

sage: V, fr, to = L.absolute_vector_space()

sage: fr(V([1,3,0,1/17])) # indirect doctest

33/17*a - 37/17*b + 1

sage: fr(to(a)), fr(to(b)) # indirect doctest

(a, b)

"""

return self.__from_K(self.__from_V(x))

class MapRelativeNumberFieldToVectorSpace(NumberFieldIsomorphism):

r"""

The isomorphism from a relative number field to its underlying `\QQ`-vector

space. Compare :class:`~MapRelativeNumberFieldToRelativeVectorSpace`.

EXAMPLES::

sage: K.<a> = NumberField(x^8 + 100*x^6 + x^2 + 5)

sage: L = K.relativize(K.subfields(4)[0][1], 'b'); L

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

sage: L_to_K, K_to_L = L.structure()

sage: V, fr, to = L.absolute_vector_space()

sage: V

Vector space of dimension 8 over Rational Field

sage: fr

Isomorphism map:

From: Vector space of dimension 8 over Rational Field

To: Number Field in b with defining polynomial x^2 + a0 over its base field

sage: to

Isomorphism map:

From: Number Field in b with defining polynomial x^2 + a0 over its base field

To: Vector space of dimension 8 over Rational Field

sage: type(fr), type(to)

(<class 'sage.rings.number_field.maps.MapVectorSpaceToRelativeNumberField'>,

<class 'sage.rings.number_field.maps.MapRelativeNumberFieldToVectorSpace'>)

sage: v = V([1, 1, 1, 1, 0, 1, 1, 1])

sage: fr(v), to(fr(v)) == v

((-a0^3 + a0^2 - a0 + 1)*b - a0^3 - a0 + 1, True)

sage: to(L.gen()), fr(to(L.gen())) == L.gen()

((0, 1, 0, 0, 0, 0, 0, 0), True)

"""

def __init__(self, L, V, to_K, to_V):

r"""

EXAMPLES::

sage: L.<a, b> = NumberField([x^2 + 3, x^2 + 5])

sage: V, fr, to = L.absolute_vector_space() # indirect doctest

sage: to

Isomorphism map:

From: Number Field in a with defining polynomial x^2 + 3 over its base field

To: Vector space of dimension 4 over Rational Field

"""

self.__to_K = to_K

self.__to_V = to_V

NumberFieldIsomorphism.__init__(self, Hom(L, V))

def _call_(self, x):

r"""

EXAMPLES::

sage: L.<a, b> = NumberField([x^2 + 3, x^2 + 5])

sage: V, fr, to = L.absolute_vector_space()

sage: to(1 + 2*a + 3*b + 4*a*b) # indirect doctest

(-15, -41/2, -2, -5/4)

sage: to(fr(V([1,3,0,1/17]))) # indirect doctest

(1, 3, 0, 1/17)

"""

return self.__to_V(self.__to_K(x))

Coverage for local/lib/python2.7/site-packages/sage/rings/number_field/maps.py : 56%

91 statements 51 run 40 missing 0 excluded