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2166

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2168

2169

2170

2171

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2173

2174

2175

""" 

Orders in Number Fields 

 

AUTHORS: 

 

- William Stein and Robert Bradshaw (2007-09): initial version 

 

EXAMPLES: 

 

We define an absolute order:: 

 

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

sage: O.basis() 

[1, 2*a] 

 

We compute a basis for an order in a relative extension 

that is generated by 2 elements:: 

 

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

sage: O.basis() 

[1, 3*a - 2*b, -6*b*a + 6, 3*a] 

 

We compute a maximal order of a degree 10 field:: 

 

sage: K.<a> = NumberField((x+1)^10 + 17) 

sage: K.maximal_order() 

Maximal Order in Number Field in a with defining polynomial x^10 + 10*x^9 + 45*x^8 + 120*x^7 + 210*x^6 + 252*x^5 + 210*x^4 + 120*x^3 + 45*x^2 + 10*x + 18 

 

We compute a suborder, which has index a power of 17 in the maximal order:: 

 

sage: O = K.order(17*a); O 

Order in Number Field in a with defining polynomial x^10 + 10*x^9 + 45*x^8 + 120*x^7 + 210*x^6 + 252*x^5 + 210*x^4 + 120*x^3 + 45*x^2 + 10*x + 18 

sage: m = O.index_in(K.maximal_order()); m 

23453165165327788911665591944416226304630809183732482257 

sage: factor(m) 

17^45 

""" 

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

# This program is free software: you can redistribute it and/or modify 

# it under the terms of the GNU General Public License as published by 

# the Free Software Foundation, either version 2 of the License, or 

# (at your option) any later version. 

# http://www.gnu.org/licenses/ 

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

from __future__ import absolute_import 

 

import six 

 

from sage.misc.cachefunc import cached_method 

from sage.rings.ring import IntegralDomain 

from sage.structure.sequence import Sequence 

from sage.rings.integer_ring import ZZ 

from sage.structure.element import is_Element 

 

from .number_field_element import OrderElement_absolute, OrderElement_relative 

 

from .number_field_element_quadratic import OrderElement_quadratic 

 

from sage.rings.monomials import monomials 

 

from sage.libs.pari.all import pari 

 

 

def is_NumberFieldOrder(R): 

r""" 

Return True if R is either an order in a number field or is the ring `\ZZ` of integers. 

 

EXAMPLES:: 

 

sage: from sage.rings.number_field.order import is_NumberFieldOrder 

sage: is_NumberFieldOrder(NumberField(x^2+1,'a').maximal_order()) 

True 

sage: is_NumberFieldOrder(ZZ) 

True 

sage: is_NumberFieldOrder(QQ) 

False 

sage: is_NumberFieldOrder(45) 

False 

""" 

return isinstance(R, Order) or R == ZZ 

 

 

def EquationOrder(f, names, **kwds): 

r""" 

Return the equation order generated by a root of the irreducible 

polynomial f or list of polynomials `f` (to construct a relative 

equation order). 

 

IMPORTANT: Note that the generators of the returned order need 

*not* be roots of `f`, since the generators of an order are -- in 

Sage -- module generators. 

 

EXAMPLES:: 

 

sage: O.<a,b> = EquationOrder([x^2+1, x^2+2]) 

sage: O 

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

sage: O.0 

-b*a - 1 

sage: O.1 

-3*a + 2*b 

 

Of course the input polynomial must be integral:: 

 

sage: R = EquationOrder(x^3 + x + 1/3, 'alpha'); R 

Traceback (most recent call last): 

... 

ValueError: each generator must be integral 

 

sage: R = EquationOrder( [x^3 + x + 1, x^2 + 1/2], 'alpha'); R 

Traceback (most recent call last): 

... 

ValueError: each generator must be integral 

""" 

from .number_field import NumberField 

R = ZZ['x'] 

if isinstance(f, (list, tuple)): 

for g in f: 

try: 

R(g) 

except TypeError: 

raise ValueError('each generator must be integral') 

else: 

try: 

R(f) 

except TypeError: 

raise ValueError('each generator must be integral') 

 

K = NumberField(f, names=names, **kwds) 

return K.order(K.gens()) 

 

 

class Order(IntegralDomain): 

r""" 

An order in a number field. 

 

An order is a subring of the number field that has `\ZZ`-rank equal 

to the degree of the number field over `\QQ`. 

 

EXAMPLES:: 

 

sage: K.<theta> = NumberField(x^4 + x + 17) 

sage: K.maximal_order() 

Maximal Order in Number Field in theta with defining polynomial x^4 + x + 17 

sage: R = K.order(17*theta); R 

Order in Number Field in theta with defining polynomial x^4 + x + 17 

sage: R.basis() 

[1, 17*theta, 289*theta^2, 4913*theta^3] 

sage: R = K.order(17*theta, 13*theta); R 

Order in Number Field in theta with defining polynomial x^4 + x + 17 

sage: R.basis() 

[1, theta, theta^2, theta^3] 

sage: R = K.order([34*theta, 17*theta + 17]); R 

Order in Number Field in theta with defining polynomial x^4 + x + 17 

 

sage: K.<b> = NumberField(x^4 + x^2 + 2) 

sage: (b^2).charpoly().factor() 

(x^2 + x + 2)^2 

sage: K.order(b^2) 

Traceback (most recent call last): 

... 

ValueError: the rank of the span of gens is wrong 

""" 

def __init__(self, K, is_maximal): 

""" 

This is called when creating an order to set the ambient field. 

 

EXAMPLES:: 

 

sage: k = CyclotomicField(5) 

sage: k.maximal_order() 

Maximal Order in Cyclotomic Field of order 5 and degree 4 

 

TESTS:: 

 

sage: k.<alg> = NumberField(x^7+3*x+1, embedding=CC(0,1)) 

sage: O = k.order(alg) 

sage: ordelt = O(alg) 

sage: CC(ordelt) 

0.0535229072603327 + 1.20934552493846*I 

""" 

self._K = K 

self._is_maximal = is_maximal 

IntegralDomain.__init__(self, ZZ, names = K.variable_names(), normalize = False) 

self._populate_coercion_lists_(embedding=self.number_field()) 

 

def fractional_ideal(self, *args, **kwds): 

""" 

Return the fractional ideal of the maximal order with given 

generators. 

 

EXAMPLES:: 

 

sage: K.<a> = NumberField(x^2 + 2) 

sage: R = K.maximal_order() 

sage: R.fractional_ideal(2/3 + 7*a, a) 

Fractional ideal (1/3*a) 

""" 

return self.number_field().fractional_ideal(*args, **kwds) 

 

def ideal(self, *args, **kwds): 

""" 

Return the integral ideal with given generators. 

 

EXAMPLES:: 

 

sage: K.<a> = NumberField(x^2 + 7) 

sage: R = K.maximal_order() 

sage: R.ideal(2/3 + 7*a, a) 

Traceback (most recent call last): 

... 

ValueError: ideal must be integral; use fractional_ideal to create a non-integral ideal. 

sage: R.ideal(7*a, 77 + 28*a) 

Fractional ideal (7) 

sage: R = K.order(4*a) 

sage: R.ideal(8) 

Traceback (most recent call last): 

... 

NotImplementedError: ideals of non-maximal orders not yet supported. 

 

This function is called implicitly below:: 

 

sage: R = EquationOrder(x^2 + 2, 'a'); R 

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

sage: (3,15)*R 

Fractional ideal (3) 

 

The zero ideal is handled properly:: 

 

sage: R.ideal(0) 

Ideal (0) of Number Field in a with defining polynomial x^2 + 2 

""" 

if not self.is_maximal(): 

raise NotImplementedError("ideals of non-maximal orders not yet supported.") 

I = self.number_field().ideal(*args, **kwds) 

if not I.is_integral(): 

raise ValueError("ideal must be integral; use fractional_ideal to create a non-integral ideal.") 

return I 

 

def _coerce_map_from_(self, R): 

""" 

Orders currently only have coerce maps from the integers. 

 

EXAMPLES:: 

 

sage: k.<a> = NumberField(x^2 + 5077) 

sage: Ok = k.maximal_order() 

sage: Ok.has_coerce_map_from(k) #indirect doctest 

False 

sage: Ok.has_coerce_map_from(ZZ) 

True 

""" 

return R is ZZ or R in six.integer_types 

 

def __mul__(self, right): 

""" 

Create an ideal in this order using the notation ``Ok*gens`` 

 

EXAMPLES:: 

 

sage: k.<a> = NumberField(x^2 + 5077); G = k.class_group(); G 

Class group of order 22 with structure C22 of Number Field in a with defining polynomial x^2 + 5077 

sage: G.0 ^ -9 

Fractional ideal class (11, a + 7) 

sage: Ok = k.maximal_order(); Ok 

Maximal Order in Number Field in a with defining polynomial x^2 + 5077 

sage: Ok * (11, a + 7) 

Fractional ideal (11, a + 7) 

sage: (11, a + 7) * Ok 

Fractional ideal (11, a + 7) 

""" 

if self.is_maximal(): 

return self._K.ideal(right) 

raise TypeError 

 

def __rmul__(self, left): 

""" 

Create an ideal in this order using the notation ``gens*Ok``. 

 

EXAMPLES:: 

 

sage: k.<a> = NumberField(x^2 + 431); G = k.class_group(); G 

Class group of order 21 with structure C21 of Number Field in a with defining polynomial x^2 + 431 

sage: G.0 # random output 

Fractional ideal class (6, 1/2*a + 11/2) 

sage: Ok = k.maximal_order(); Ok 

Maximal Order in Number Field in a with defining polynomial x^2 + 431 

sage: (6, 1/2*a + 11/2)*Ok # random output 

Fractional ideal (6, 1/2*a + 11/2) 

sage: 17*Ok 

Fractional ideal (17) 

""" 

return self * left 

 

def is_maximal(self): 

""" 

Return ``True`` if this is the maximal order. 

 

EXAMPLES:: 

 

sage: k.<i> = NumberField(x^2 + 1) 

sage: O3 = k.order(3*i); O5 = k.order(5*i); Ok = k.maximal_order(); Osum = O3 + O5 

sage: Osum.is_maximal() 

True 

sage: O3.is_maximal() 

False 

sage: O5.is_maximal() 

False 

sage: Ok.is_maximal() 

True 

 

An example involving a relative order:: 

 

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

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

sage: O.is_maximal() 

False 

 

""" 

if self._is_maximal is None: 

self._is_maximal = (self.absolute_discriminant() == self._K.absolute_discriminant()) 

return self._is_maximal 

 

def is_field(self, proof = True): 

r""" 

Return ``False`` (because an order is never a field). 

 

EXAMPLES:: 

 

sage: L.<alpha> = NumberField(x**4 - x**2 + 7) 

sage: O = L.maximal_order() ; O.is_field() 

False 

sage: CyclotomicField(12).ring_of_integers().is_field() 

False 

""" 

return False 

 

def is_noetherian(self): 

r""" 

Return ``True`` (because orders are always Noetherian) 

 

EXAMPLES:: 

 

sage: L.<alpha> = NumberField(x**4 - x**2 + 7) 

sage: O = L.maximal_order() ; O.is_noetherian() 

True 

sage: E.<w> = NumberField(x^2 - x + 2) 

sage: OE = E.ring_of_integers(); OE.is_noetherian() 

True 

""" 

return True 

 

def is_integrally_closed(self): 

""" 

Return ``True`` if this ring is integrally closed, i.e., is equal 

to the maximal order. 

 

EXAMPLES:: 

 

sage: K.<a> = NumberField(x^2 + 189*x + 394) 

sage: R = K.order(2*a) 

sage: R.is_integrally_closed() 

False 

sage: R 

Order in Number Field in a with defining polynomial x^2 + 189*x + 394 

sage: S = K.maximal_order(); S 

Maximal Order in Number Field in a with defining polynomial x^2 + 189*x + 394 

sage: S.is_integrally_closed() 

True 

""" 

return self.is_maximal() 

 

def krull_dimension(self): 

""" 

Return the Krull dimension of this order, which is 1. 

 

EXAMPLES:: 

 

sage: K.<a> = QuadraticField(5) 

sage: OK = K.maximal_order() 

sage: OK.krull_dimension() 

1 

sage: O2 = K.order(2*a) 

sage: O2.krull_dimension() 

1 

""" 

return ZZ(1) 

 

def integral_closure(self): 

""" 

Return the integral closure of this order. 

 

EXAMPLES:: 

 

sage: K.<a> = QuadraticField(5) 

sage: O2 = K.order(2*a); O2 

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

sage: O2.integral_closure() 

Maximal Order in Number Field in a with defining polynomial x^2 - 5 

sage: OK = K.maximal_order() 

sage: OK is OK.integral_closure() 

True 

""" 

if self.is_maximal(): 

return self 

else: 

return self.number_field().maximal_order() 

 

def gen(self, i): 

r""" 

Return `i`'th module generator of this order. 

 

EXAMPLES:: 

 

sage: K.<c> = NumberField(x^3 + 2*x + 17) 

sage: O = K.maximal_order(); O 

Maximal Order in Number Field in c with defining polynomial x^3 + 2*x + 17 

sage: O.basis() 

[1, c, c^2] 

sage: O.gen(1) 

c 

sage: O.gen(2) 

c^2 

sage: O.gen(5) 

Traceback (most recent call last): 

... 

IndexError: no 5th generator 

sage: O.gen(-1) 

Traceback (most recent call last): 

... 

IndexError: no -1th generator 

""" 

b = self.basis() 

if i < 0 or i >= len(b): 

raise IndexError("no %sth generator" % i) 

return self.basis()[i] 

 

def ngens(self): 

""" 

Return the number of module generators of this order. 

 

EXAMPLES:: 

 

sage: K.<a> = NumberField(x^3 + x^2 - 2*x + 8) 

sage: O = K.maximal_order() 

sage: O.ngens() 

3 

""" 

return self.absolute_degree() 

 

def basis(self): # this must be defined in derived class 

r""" 

Return a basis over `\ZZ` of this order. 

 

EXAMPLES:: 

 

sage: K.<a> = NumberField(x^3 + x^2 - 16*x + 16) 

sage: O = K.maximal_order(); O 

Maximal Order in Number Field in a with defining polynomial x^3 + x^2 - 16*x + 16 

sage: O.basis() 

[1, 1/4*a^2 + 1/4*a, a^2] 

""" 

raise NotImplementedError 

 

def coordinates(self, x): 

r""" 

Return the coordinate vector of `x` with respect to this order. 

 

INPUT: 

 

- ``x`` -- an element of the number field of this order. 

 

OUTPUT: 

 

A vector of length `n` (the degree of the field) giving 

the coordinates of `x` with respect to the integral basis 

of the order. In general this will be a vector of 

rationals; it will consist of integers if and only if `x` 

is in the order. 

 

AUTHOR: John Cremona 2008-11-15 

 

ALGORITHM: 

 

Uses linear algebra. The change-of-basis matrix is 

cached. Provides simpler implementations for 

``_contains_()``, ``is_integral()`` and ``smallest_integer()``. 

 

EXAMPLES:: 

 

sage: K.<i> = QuadraticField(-1) 

sage: OK = K.ring_of_integers() 

sage: OK_basis = OK.basis(); OK_basis 

[1, i] 

sage: a = 23-14*i 

sage: acoords = OK.coordinates(a); acoords 

(23, -14) 

sage: sum([OK_basis[j]*acoords[j] for j in range(2)]) == a 

True 

sage: OK.coordinates((120+340*i)/8) 

(15, 85/2) 

 

sage: O = K.order(3*i) 

sage: O.is_maximal() 

False 

sage: O.index_in(OK) 

3 

sage: acoords = O.coordinates(a); acoords 

(23, -14/3) 

sage: sum([O.basis()[j]*acoords[j] for j in range(2)]) == a 

True 

 

""" 

K = self.number_field() 

V, from_V, to_V = K.absolute_vector_space() 

 

try: 

M = self.__basis_matrix_inverse 

except AttributeError: 

from sage.matrix.constructor import Matrix 

self.__basis_matrix_inverse = Matrix([to_V(b) for b in self.basis()]).inverse() 

M = self.__basis_matrix_inverse 

return to_V(K(x))*M 

 

def free_module(self): 

r""" 

Return the free `\ZZ`-module contained in the vector space 

associated to the ambient number field, that corresponds 

to this ideal. 

 

EXAMPLES:: 

 

sage: K.<a> = NumberField(x^3 + x^2 - 2*x + 8) 

sage: O = K.maximal_order(); O.basis() 

[1, 1/2*a^2 + 1/2*a, a^2] 

sage: O.free_module() 

Free module of degree 3 and rank 3 over Integer Ring 

User basis matrix: 

[ 1 0 0] 

[ 0 1/2 1/2] 

[ 0 0 1] 

 

An example in a relative extension. Notice that the module is 

a `\ZZ`-module in the absolute_field associated to the relative 

field:: 

 

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

sage: O = K.maximal_order(); O.basis() 

[(-3/2*b - 5)*a + 7/2*b - 2, -3*a + 2*b, -2*b*a - 3, -7*a + 5*b] 

sage: O.free_module() 

Free module of degree 4 and rank 4 over Integer Ring 

User basis matrix: 

[1/4 1/4 3/4 3/4] 

[ 0 1/2 0 1/2] 

[ 0 0 1 0] 

[ 0 0 0 1] 

""" 

try: 

return self.__free_module 

except AttributeError: 

pass 

from .number_field_ideal import basis_to_module 

M = basis_to_module(self.basis(), self.number_field()) 

self.__free_module = M 

return M 

 

@cached_method 

def ring_generators(self): 

""" 

Return generators for self as a ring. 

 

EXAMPLES:: 

 

sage: K.<i> = NumberField(x^2 + 1) 

sage: O = K.maximal_order(); O 

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

sage: O.ring_generators() 

[i] 

 

This is an example where 2 generators are required (because 2 is an essential 

discriminant divisor).:: 

 

sage: K.<a> = NumberField(x^3 + x^2 - 2*x + 8) 

sage: O = K.maximal_order(); O.basis() 

[1, 1/2*a^2 + 1/2*a, a^2] 

sage: O.ring_generators() 

[1/2*a^2 + 1/2*a, a^2] 

 

An example in a relative number field:: 

 

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

sage: O = K.maximal_order() 

sage: O.ring_generators() 

[(-5/3*b^2 + 3*b - 2)*a - 7/3*b^2 + b + 3, (-5*b^2 - 9)*a - 5*b^2 - b, (-6*b^2 - 11)*a - 6*b^2 - b] 

""" 

K = self._K 

n = [] 

V, from_V, to_V = self._K.absolute_vector_space() 

A = ZZ**K.absolute_degree() 

remaining = [x for x in self.basis() if x != 1] 

gens = [] 

while remaining: 

g = remaining.pop(0) 

gens.append(g) 

n.append(g.absolute_minpoly().degree()) 

W = A.span([to_V(x) for x in monomials(gens, n)]) 

remaining = [x for x in remaining if not to_V(x) in W] 

return Sequence(gens,immutable=True) 

 

@cached_method 

def _defining_names(self): 

""" 

Return the generators of the ambient number field, but with 

this order as parent. 

 

EXAMPLES:: 

 

sage: B.<z> = EquationOrder(x^2 + 3) 

sage: B._defining_names() 

(z,) 

 

For relative extensions:: 

 

sage: O.<a,b> = EquationOrder([x^2 + 1, x^2 + 2]) 

sage: O._defining_names() 

(a, b) 

""" 

gens = self.number_field().gens() 

return tuple(self(g) for g in gens) 

 

def zeta(self, n=2, all=False): 

r""" 

Return a primitive n-th root of unity in this order, if it 

contains one. If all is True, return all of them. 

 

EXAMPLES:: 

 

sage: F.<alpha> = NumberField(x**2+3) 

sage: F.ring_of_integers().zeta(6) 

1/2*alpha + 1/2 

sage: O = F.order([3*alpha]) 

sage: O.zeta(3) 

Traceback (most recent call last): 

... 

ArithmeticError: There are no 3rd roots of unity in self. 

""" 

roots_in_field = self.number_field().zeta(n, True) 

roots_in_self = [ self(x) for x in roots_in_field if x in self ] 

if len(roots_in_self) == 0: 

if all: 

return [] 

else: 

raise ArithmeticError("There are no %s roots of unity in self."%n.ordinal_str()) 

if all: 

return roots_in_self 

else: 

return roots_in_self[0] 

 

 

def number_field(self): 

""" 

Return the number field of this order, which is the ambient 

number field that this order is embedded in. 

 

EXAMPLES:: 

 

sage: K.<b> = NumberField(x^4 + x^2 + 2) 

sage: O = K.order(2*b); O 

Order in Number Field in b with defining polynomial x^4 + x^2 + 2 

sage: O.basis() 

[1, 2*b, 4*b^2, 8*b^3] 

sage: O.number_field() 

Number Field in b with defining polynomial x^4 + x^2 + 2 

sage: O.number_field() is K 

True 

""" 

return self._K 

 

def ambient(self): 

r""" 

Return the ambient number field that contains self. 

 

This is the same as ``self.number_field()`` and 

``self.fraction_field()`` 

 

EXAMPLES:: 

 

sage: k.<z> = NumberField(x^2 - 389) 

sage: o = k.order(389*z + 1) 

sage: o 

Order in Number Field in z with defining polynomial x^2 - 389 

sage: o.basis() 

[1, 389*z] 

sage: o.ambient() 

Number Field in z with defining polynomial x^2 - 389 

""" 

return self._K 

 

def residue_field(self, prime, names=None, check=False): 

""" 

Return the residue field of this order at a given prime, ie `O/pO`. 

 

INPUT: 

 

- ``prime`` -- a prime ideal of the maximal order in this number field. 

- ``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: OK = K.maximal_order() 

sage: OK.residue_field(P) 

Residue field in abar of Fractional ideal (61, a^2 + 30) 

sage: Fp.<b> = OK.residue_field(P) 

sage: Fp 

Residue field in b of Fractional ideal (61, a^2 + 30) 

""" 

if self.is_maximal(): 

return self.number_field().residue_field(prime, names, check) 

 

raise NotImplementedError("Residue fields of non-maximal orders " 

"are not yet supported.") 

 

def fraction_field(self): 

""" 

Return the fraction field of this order, which is the 

ambient number field. 

 

EXAMPLES:: 

 

sage: K.<b> = NumberField(x^4 + 17*x^2 + 17) 

sage: O = K.order(17*b); O 

Order in Number Field in b with defining polynomial x^4 + 17*x^2 + 17 

sage: O.fraction_field() 

Number Field in b with defining polynomial x^4 + 17*x^2 + 17 

""" 

return self._K 

 

def degree(self): 

r""" 

Return the degree of this order, which is the rank of this order as a 

`\ZZ`-module. 

 

EXAMPLES:: 

 

sage: k.<c> = NumberField(x^3 + x^2 - 2*x+8) 

sage: o = k.maximal_order() 

sage: o.degree() 

3 

sage: o.rank() 

3 

""" 

return self._K.degree() 

 

def rank(self): 

r""" 

Return the rank of this order, which is the rank of the underlying 

`\ZZ`-module, or the degree of the ambient number field that contains 

this order. 

 

This is a synonym for ``degree()``. 

 

EXAMPLES:: 

 

sage: k.<c> = NumberField(x^5 + x^2 + 1) 

sage: o = k.maximal_order(); o 

Maximal Order in Number Field in c with defining polynomial x^5 + x^2 + 1 

sage: o.rank() 

5 

""" 

return self.degree() 

 

def class_number(self, proof=None): 

r""" 

Return the class number of this order. 

 

EXAMPLES:: 

 

sage: ZZ[2^(1/3)].class_number() 

1 

sage: QQ[sqrt(-23)].maximal_order().class_number() 

3 

sage: ZZ[120*sqrt(-23)].class_number() 

288 

 

Note that non-maximal orders are only supported in quadratic fields:: 

 

sage: ZZ[120*sqrt(-23)].class_number() 

288 

sage: ZZ[100*sqrt(3)].class_number() 

4 

sage: ZZ[11*2^(1/3)].class_number() 

Traceback (most recent call last): 

... 

NotImplementedError: computation of class numbers of non-maximal orders not in quadratic fields is not implemented 

 

""" 

if not self.is_maximal(): 

K = self.number_field() 

if K.degree() != 2: 

raise NotImplementedError("computation of class numbers of non-maximal orders not in quadratic fields is not implemented") 

return ZZ(pari.qfbclassno(self.discriminant())) 

return self.number_field().class_number(proof=proof) 

 

def class_group(self, proof=None, names='c'): 

r""" 

Return the class group of this order. 

 

(Currently only implemented for the maximal order.) 

 

EXAMPLES:: 

 

sage: k.<a> = NumberField(x^2 + 5077) 

sage: O = k.maximal_order(); O 

Maximal Order in Number Field in a with defining polynomial x^2 + 5077 

sage: O.class_group() 

Class group of order 22 with structure C22 of Number Field in a with defining polynomial x^2 + 5077 

""" 

if self.is_maximal(): 

return self.number_field().class_group(proof=proof, names=names) 

else: 

raise NotImplementedError 

 

def is_suborder(self, other): 

""" 

Return True if self and other are both orders in the 

same ambient number field and self is a subset of other. 

 

EXAMPLES:: 

 

sage: W.<i> = NumberField(x^2 + 1) 

sage: O5 = W.order(5*i) 

sage: O10 = W.order(10*i) 

sage: O15 = W.order(15*i) 

sage: O15.is_suborder(O5) 

True 

sage: O5.is_suborder(O15) 

False 

sage: O10.is_suborder(O15) 

False 

 

We create another isomorphic but different field:: 

 

sage: W2.<j> = NumberField(x^2 + 1) 

sage: P5 = W2.order(5*j) 

 

This is False because the ambient number fields are not equal.:: 

 

sage: O5.is_suborder(P5) 

False 

 

We create a field that contains (in no natural way!) W, 

and of course again is_suborder returns False:: 

 

sage: K.<z> = NumberField(x^4 + 1) 

sage: M = K.order(5*z) 

sage: O5.is_suborder(M) 

False 

""" 

if not isinstance(other, Order): 

return False 

if other.number_field() != self.number_field(): 

return False 

return self.module().is_submodule(other.module()) 

 

def __eq__(self, other): 

r""" 

Check whether the order ``self`` is equal to ``other``. 

 

.. NOTE:: 

 

This method is just for equality. If you want to check if 

``self`` is contained in ``other``, use instead 

``self.is_suborder(other)`` to determine inclusion. 

 

EXAMPLES:: 

 

sage: K.<a> = NumberField(x^3 + 2) 

sage: O1 = K.order(a); O1 

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

sage: O2 = K.order(a^2); O2 

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

sage: O1 == O2 

False 

 

sage: O1 == K 

False 

sage: K == O1 

False 

 

Here is how to check for inclusion:: 

 

sage: O2.is_suborder(O1) 

True 

""" 

if not isinstance(other, Order): 

return False 

if self._K != other._K: 

return False 

if self is other: 

return True 

return self._module_rep == other._module_rep 

 

def __ne__(self, other): 

""" 

Check whether the order ``self`` is not equal to ``other``. 

 

EXAMPLES:: 

 

sage: K.<a> = NumberField(x^3 + 2) 

sage: O1 = K.order(a); O1 

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

sage: O2 = K.order(a^2); O2 

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

sage: O1 != O2 

True 

""" 

return not (self == other) 

 

def random_element(self, *args, **kwds): 

""" 

Return a random element of this order. 

 

INPUT: 

 

- ``args``, ``kwds`` -- parameters passed to the random 

integer function. See the documentation for 

``ZZ.random_element()`` for details. 

 

OUTPUT: 

 

A random element of this order, computed as a random 

`\ZZ`-linear combination of the basis. 

 

EXAMPLES:: 

 

sage: K.<a> = NumberField(x^3 + 2) 

sage: OK = K.ring_of_integers() 

sage: OK.random_element() # random output 

-2*a^2 - a - 2 

sage: OK.random_element(distribution="uniform") # random output 

-a^2 - 1 

sage: OK.random_element(-10,10) # random output 

-10*a^2 - 9*a - 2 

sage: K.order(a).random_element() # random output 

a^2 - a - 3 

 

:: 

 

sage: K.<z> = CyclotomicField(17) 

sage: OK = K.ring_of_integers() 

sage: OK.random_element() # random output 

z^15 - z^11 - z^10 - 4*z^9 + z^8 + 2*z^7 + z^6 - 2*z^5 - z^4 - 445*z^3 - 2*z^2 - 15*z - 2 

sage: OK.random_element().is_integral() 

True 

sage: OK.random_element().parent() is OK 

True 

 

A relative example:: 

 

sage: K.<a, b> = NumberField([x^2 + 2, x^2 + 1000*x + 1]) 

sage: OK = K.ring_of_integers() 

sage: OK.random_element() # random output 

(42221/2*b + 61/2)*a + 7037384*b + 7041 

sage: OK.random_element().is_integral() # random output 

True 

sage: OK.random_element().parent() is OK # random output 

True 

 

An example in a non-maximal order:: 

 

sage: K.<a> = QuadraticField(-3) 

sage: R = K.ring_of_integers() 

sage: A = K.order(a) 

sage: A.index_in(R) 

2 

sage: R.random_element() # random output 

-39/2*a - 1/2 

sage: A.random_element() # random output 

2*a - 1 

sage: A.random_element().is_integral() 

True 

sage: A.random_element().parent() is A 

True 

""" 

return sum([ZZ.random_element(*args, **kwds)*a for a in self.basis()]) 

 

def absolute_degree(self): 

r""" 

Returns the absolute degree of this order, ie the degree of this order over `\ZZ`. 

 

EXAMPLES:: 

 

sage: K.<a> = NumberField(x^3 + 2) 

sage: O = K.maximal_order() 

sage: O.absolute_degree() 

3 

""" 

return self.number_field().absolute_degree() 

 

def valuation(self, p): 

r""" 

Return the ``p``-adic valuation on this order. 

 

EXAMPLES: 

 

The valuation can be specified with an integer ``prime`` that is 

completely ramified or unramified:: 

 

sage: K.<a> = NumberField(x^2 + 1) 

sage: O = K.order(2*a) 

sage: valuations.pAdicValuation(O, 2) 

2-adic valuation 

 

sage: GaussianIntegers().valuation(2) 

2-adic valuation 

 

:: 

 

sage: GaussianIntegers().valuation(3) 

3-adic valuation 

 

A ``prime`` that factors into pairwise distinct factors, results in an error:: 

 

sage: GaussianIntegers().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).ring_of_integers().valuation(ZZ.valuation(5)) 

5-adic valuation 

 

When the extension is not unique, this does not work:: 

 

sage: GaussianIntegers().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 

 

If the fraction field 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:: 

 

sage: R.<x> = QQ[] 

sage: v = GaussianIntegers().valuation(GaussValuation(R, QQ.valuation(5)).augmentation(x + 2, infinity)) 

sage: w = GaussianIntegers().valuation(GaussValuation(R, QQ.valuation(5)).augmentation(x + 1/2, infinity)) 

sage: v == w 

False 

 

.. SEEALSO:: 

 

:meth:`NumberField_generic.valuation() <sage.rings.number_field.number_field.NumberField_generic.valuation>`, 

:meth:`pAdicGeneric.valuation() <sage.rings.padics.padic_generic.pAdicGeneric.valuation>` 

 

""" 

from sage.rings.padics.padic_valuation import pAdicValuation 

return pAdicValuation(self, p) 

 

def some_elements(self): 

""" 

Return a list of elements of the given order. 

 

EXAMPLES:: 

 

sage: G = GaussianIntegers(); G 

Gaussian Integers in Number Field in I with defining polynomial x^2 + 1 

sage: G.some_elements() 

[1, I, 2*I, -1, 0, -I, 2, 4*I, -2, -2*I, -4] 

 

sage: R.<t> = QQ[] 

sage: K.<a> = QQ.extension(t^3 - 2); K 

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

sage: Z = K.ring_of_integers(); Z 

Maximal Order in Number Field in a with defining polynomial t^3 - 2 

sage: Z.some_elements() 

[1, a, a^2, 2*a, 0, 2, a^2 + 2*a + 1, ..., a^2 + 1, 2*a^2 + 2, a^2 + 2*a, 4*a^2 + 4] 

 

TESTS: 

 

This also works for trivial extensions:: 

 

sage: R.<t> = QQ[] 

sage: K.<a> = QQ.extension(t); K 

Number Field in a with defining polynomial t 

sage: Z = K.ring_of_integers(); Z 

Maximal Order in Number Field in a with defining polynomial t 

sage: Z.some_elements() 

[1, 0, 2, -1, -2, 4] 

 

""" 

elements = list(self.basis()) 

for a in self.fraction_field().some_elements(): 

if a in self and a not in elements: 

elements.append(self(a)) 

return elements 

 

## def absolute_polynomial(self): 

## """ 

## Returns the absolute polynomial of this order, which is just the absolute polynomial of the number field. 

 

## EXAMPLES:: 

 

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

## Traceback (most recent call last): 

## ... 

## NotImplementedError 

 

## #sage: OK.absolute_polynomial() 

## #x^6 + 5*x^4 - 2*x^3 + 4*x^2 + 4*x + 1 

## """ 

## return self.number_field().absolute_polynomial() 

 

## def polynomial(self): 

## """ 

## Returns the polynomial defining the number field that contains self. 

## """ 

## return self.number_field().polynomial() 

 

## def polynomial_ntl(self): 

## """ 

## Return defining polynomial of the parent 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 + 1,'a').maximal_order().polynomial_ntl() 

## ([1 0 1], 1) 

## """ 

## return self.number_field().polynomial_ntl() 

 

class AbsoluteOrder(Order): 

 

def __init__(self, K, module_rep, is_maximal=None, check=True): 

""" 

EXAMPLES:: 

 

sage: from sage.rings.number_field.order import * 

sage: x = polygen(QQ) 

sage: K.<a> = NumberField(x^3+2) 

sage: V, from_v, to_v = K.vector_space() 

sage: M = span([to_v(a^2), to_v(a), to_v(1)],ZZ) 

sage: O = AbsoluteOrder(K, M); O 

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

 

sage: M = span([to_v(a^2), to_v(a), to_v(2)],ZZ) 

sage: O = AbsoluteOrder(K, M); O 

Traceback (most recent call last): 

... 

ValueError: 1 is not in the span of the module, hence not an order. 

 

sage: loads(dumps(O)) == O 

True 

 

Quadratic elements have a special optimized type: 

 

""" 

if K.degree() == 2: 

self._element_type = OrderElement_quadratic 

# adding the following attribute makes the comparison of elements 

# faster. 

self._standard_embedding = K._standard_embedding 

else: 

self._element_type = OrderElement_absolute 

 

self._module_rep = module_rep 

V, from_v, to_v = K.vector_space() 

Order.__init__(self, K, is_maximal=is_maximal) 

 

if check: 

if not K.is_absolute(): 

raise ValueError("AbsoluteOrder must be called with an absolute number field.") 

if to_v(1) not in module_rep: 

raise ValueError("1 is not in the span of the module, hence not an order.") 

if module_rep.rank() != self._K.degree(): 

raise ValueError("the module must have full rank.") 

 

def _element_constructor_(self, x): 

r""" 

Coerce ``x`` into this order. 

 

EXAMPLES:: 

 

sage: x = polygen(QQ) 

sage: k.<z> = NumberField(x^2 - 389) 

sage: m = k.order(3*z); m 

Order in Number Field in z with defining polynomial x^2 - 389 

sage: m(6*z) 

6*z 

sage: k(m(6*z)) 

6*z 

 

If ``x`` is a list or tuple the element constructed is the 

linear combination of the generators with these coefficients 

(see :trac:`10017`):: 

 

sage: x = polygen(QQ) 

sage: K.<a> = NumberField(x^3-10) 

sage: ZK = K.ring_of_integers() 

sage: ZK.basis() 

[1/3*a^2 + 1/3*a + 1/3, a, a^2] 

sage: ZK([1,2,3]) 

10/3*a^2 + 7/3*a + 1/3 

sage: K([1,2,3]) 

3*a^2 + 2*a + 1 

 

""" 

if is_Element(x) and x.parent() is self: 

return x 

if isinstance(x, (tuple, list)): 

x = sum(xi*gi for xi,gi in zip(x,self.gens())) 

if not is_Element(x) or x.parent() is not self._K: 

x = self._K(x) 

V, _, embedding = self._K.vector_space() 

if not embedding(x) in self._module_rep: 

raise TypeError("Not an element of the order.") 

return self._element_type(self, x) 

 

def __reduce__(self): 

r""" 

Used in pickling. 

 

We test that :trac:`6462` is fixed. This used to fail because 

pickling the order also pickled the cached results of the 

``basis`` call, which were elements of the order. 

 

:: 

 

sage: L.<a> = QuadraticField(-1) 

sage: OL = L.maximal_order() 

sage: _ = OL.basis() 

sage: loads(dumps(OL)) == OL 

True 

""" 

return (AbsoluteOrder, (self.number_field(), self.free_module(), self._is_maximal, False)) 

 

def __add__(left, right): 

""" 

Add two orders. 

 

EXAMPLES:: 

 

sage: K.<a> = NumberField(polygen(QQ,'z')^3 - 2) 

sage: O6 = K.order(6*a); O6 

Order in Number Field in a with defining polynomial z^3 - 2 

sage: O6.basis() 

[1, 6*a, 36*a^2] 

sage: O15 = K.order(15*a^2); O15.basis() 

[1, 450*a, 15*a^2] 

sage: R = O6 + O15; R 

Order in Number Field in a with defining polynomial z^3 - 2 

sage: R.basis() 

[1, 6*a, 3*a^2] 

""" 

if not isinstance(left, AbsoluteOrder) or not isinstance(right, AbsoluteOrder): 

raise NotImplementedError 

if left.number_field() != right.number_field(): 

raise TypeError("Number fields don't match.") 

if left._is_maximal: 

return left 

elif right._is_maximal: 

return right 

return AbsoluteOrder(left._K, left._module_rep + right._module_rep, None) 

 

def __and__(left, right): 

""" 

Intersect orders. 

 

EXAMPLES:: 

 

sage: K.<i> = QuadraticField(-1) 

sage: O3 = K.order(3*i); O5 = K.order(5*i) 

sage: R = O3 & O5; R 

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

sage: R.basis() 

[1, 15*i] 

sage: O3.intersection(O5).basis() 

[1, 15*i] 

""" 

if not isinstance(left, AbsoluteOrder) or not isinstance(right, AbsoluteOrder): 

raise NotImplementedError 

if left.number_field() != right.number_field(): 

raise TypeError("Number fields don't match.") 

return AbsoluteOrder(left._K, left._module_rep.intersection(right._module_rep), False) 

 

def _magma_init_(self, magma): 

""" 

Return Magma version of this absolute order. 

 

INPUT: 

 

- ``magma`` -- a magma interpreter 

 

OUTPUT: 

 

a MagmaElement, the magma version of this absolute order 

 

EXAMPLES:: 

 

sage: K.<a> = NumberField(x^3 + 2) 

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

Equation Order with defining polynomial x^3 + 2 over its ground order 

 

_magma_init_ was called implicitly by the above call:: 

 

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

'Order([(_sage_[...]![1, 0, 0]),(_sage_[...]![0, 1, 0]),(_sage_[...]![0, 0, 1])])' 

""" 

K = self.number_field() 

v = [K(a)._magma_init_(magma) for a in self.gens()] 

return 'Order([%s])'%(','.join(v)) 

 

def discriminant(self): 

""" 

Return the discriminant of this order. 

 

EXAMPLES:: 

 

sage: K.<a> = NumberField(x^8 + x^3 - 13*x + 26) 

sage: O = K.maximal_order() 

sage: factor(O.discriminant()) 

3 * 11 * 13^2 * 613 * 1575917857 

sage: L = K.order(13*a^2) 

sage: factor(L.discriminant()) 

3^3 * 5^2 * 11 * 13^60 * 613 * 733^2 * 1575917857 

sage: factor(L.index_in(O)) 

3 * 5 * 13^29 * 733 

sage: L.discriminant() / O.discriminant() == L.index_in(O)^2 

True 

""" 

try: 

return self.__discriminant 

except AttributeError: 

if self._is_maximal: 

D = self._K.discriminant() 

else: 

D = self._K.discriminant(self.basis()) 

self.__discriminant = D 

return D 

 

absolute_discriminant = discriminant 

 

 

def change_names(self, names): 

""" 

Return a new order isomorphic to this one in the number field with 

given variable names. 

 

EXAMPLES:: 

 

sage: R = EquationOrder(x^3 + x + 1, 'alpha'); R 

Order in Number Field in alpha with defining polynomial x^3 + x + 1 

sage: R.basis() 

[1, alpha, alpha^2] 

sage: S = R.change_names('gamma'); S 

Order in Number Field in gamma with defining polynomial x^3 + x + 1 

sage: S.basis() 

[1, gamma, gamma^2] 

""" 

K = self.number_field().change_names(names) 

_, to_K = K.structure() 

B = [to_K(a) for a in self.basis()] 

return K.order(B, check_is_integral=False, check_rank=False, allow_subfield=True) 

 

def index_in(self, other): 

""" 

Return the index of self in other. This is a lattice index, 

so it is a rational number if self isn't contained in other. 

 

INPUT: 

 

- ``other`` -- another absolute order with the same ambient number field. 

 

OUTPUT: 

 

a rational number 

 

EXAMPLES:: 

 

sage: k.<i> = NumberField(x^2 + 1) 

sage: O1 = k.order(i) 

sage: O5 = k.order(5*i) 

sage: O5.index_in(O1) 

5 

 

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 

sage: O1 = k.order(a); O1 

Order in Number Field in a with defining polynomial x^3 + x^2 - 2*x + 8 

sage: O1.index_in(o) 

2 

sage: O2 = k.order(1+2*a); O2 

Order in Number Field in a with defining polynomial x^3 + x^2 - 2*x + 8 

sage: O1.basis() 

[1, a, a^2] 

sage: O2.basis() 

[1, 2*a, 4*a^2] 

sage: o.index_in(O2) 

1/16 

""" 

if not isinstance(other, AbsoluteOrder): 

raise TypeError("other must be an absolute order.") 

if other.ambient() != self.ambient(): 

raise ValueError("other must have the same ambient number field as self.") 

return self._module_rep.index_in(other._module_rep) 

 

def module(self): 

""" 

Returns the underlying free module corresponding to this 

order, embedded in the vector space corresponding to the 

ambient number field. 

 

EXAMPLES:: 

 

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

sage: m = k.order(3*a); m 

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

sage: m.module() 

Free module of degree 3 and rank 3 over Integer Ring 

Echelon basis matrix: 

[1 0 0] 

[0 3 0] 

[0 0 9] 

""" 

return self._module_rep 

 

def intersection(self, other): 

""" 

Return the intersection of this order with another order. 

 

EXAMPLES:: 

 

sage: k.<i> = NumberField(x^2 + 1) 

sage: O6 = k.order(6*i) 

sage: O9 = k.order(9*i) 

sage: O6.basis() 

[1, 6*i] 

sage: O9.basis() 

[1, 9*i] 

sage: O6.intersection(O9).basis() 

[1, 18*i] 

sage: (O6 & O9).basis() 

[1, 18*i] 

sage: (O6 + O9).basis() 

[1, 3*i] 

""" 

return self & other 

 

def _repr_(self): 

""" 

Return print representation of this absolute order. 

 

EXAMPLES:: 

 

sage: K.<a> = NumberField(x^4 - 5) 

sage: K.maximal_order()._repr_() 

'Maximal Order in Number Field in a with defining polynomial x^4 - 5' 

sage: K.order(a)._repr_() 

'Order in Number Field in a with defining polynomial x^4 - 5' 

 

We have special cases for Gaussian and Eisenstein integers:: 

 

sage: K = CyclotomicField(4) 

sage: K.ring_of_integers() 

Gaussian Integers in Cyclotomic Field of order 4 and degree 2 

sage: K = QuadraticField(-3) 

sage: K.ring_of_integers() 

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

""" 

if self._is_maximal: 

s = "Maximal Order" 

if self.degree() == 2: 

D = self.discriminant() 

if D == -3: 

s = "Eisenstein Integers" 

if D == -4: 

s = "Gaussian Integers" 

else: 

s = "Order" 

return s + " in " + repr(self._K) 

 

def basis(self): 

r""" 

Return the basis over `\ZZ` for this order. 

 

EXAMPLES:: 

 

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

sage: O = k.maximal_order(); O 

Maximal Order in Number Field in c with defining polynomial x^3 + x^2 + 1 

sage: O.basis() 

[1, c, c^2] 

 

The basis is an immutable sequence:: 

 

sage: type(O.basis()) 

<class 'sage.structure.sequence.Sequence_generic'> 

 

The generator functionality uses the basis method:: 

 

sage: O.0 

1 

sage: O.1 

c 

sage: O.basis() 

[1, c, c^2] 

sage: O.ngens() 

3 

""" 

try: 

return self.__basis 

except AttributeError: 

V, from_V, to_V = self._K.vector_space() 

B = Sequence([self(from_V(b)) for b in self._module_rep.basis()], immutable=True) 

self.__basis = B 

return B 

 

def absolute_order(self): 

""" 

Return the absolute order associated to this order, which is 

just this order again since this is an absolute order. 

 

EXAMPLES:: 

 

sage: K.<a> = NumberField(x^3 + 2) 

sage: O1 = K.order(a); O1 

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

sage: O1.absolute_order() is O1 

True 

""" 

return self 

 

class RelativeOrder(Order): 

""" 

A relative order in a number field. 

 

A relative order is an order in some relative number field 

 

Invariants of this order may be computed with respect to the 

contained order. 

""" 

def __init__(self, K, absolute_order, is_maximal=None, check=True): 

""" 

Create the relative order. 

 

EXAMPLES:: 

 

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

sage: O = k.maximal_order(); O # indirect doctest 

Maximal Relative Order in Number Field in a with defining polynomial x^2 - 3 over its base field 

 

sage: _ = O.basis() 

sage: loads(dumps(O)) == O 

True 

""" 

self._absolute_order = absolute_order 

self._module_rep = absolute_order._module_rep 

Order.__init__(self, K, is_maximal=is_maximal) 

 

def _element_constructor_(self, x): 

""" 

Coerce an element into this relative order. 

 

EXAMPLES:: 

 

sage: K.<a, b> = NumberField([x^2 + 2, x^2 + 1000*x + 1]) 

sage: OK = K.ring_of_integers() 

sage: OK(a) 

a 

sage: OK([3, 4]) 

4*a + 3 

 

The following used to fail; see :trac:`5276`:: 

 

sage: S.<y> = OK[]; S 

Univariate Polynomial Ring in y over Maximal Relative Order in Number Field in a with defining polynomial x^2 + 2 over its base field 

 

We test that :trac:`4193` is also fixed:: 

 

sage: K1.<a> = NumberField(x^3 - 2) 

sage: R.<y> = PolynomialRing(K1) 

sage: K2.<b> = K1.extension(y^2 - a) 

sage: R = K2.order(b) 

sage: b in R 

True 

sage: bb = R.basis()[1] # b by any other name 

sage: bb == b 

True 

sage: bb.parent() is R 

True 

sage: bb in R # this used to return False 

True 

sage: R(bb) == bb # this used to raise an error 

True 

""" 

 

x = self._K(x) 

abs_order = self._absolute_order 

to_abs = abs_order._K.structure()[1] 

x = abs_order(to_abs(x)) # will test membership 

return OrderElement_relative(self, x) 

 

def _repr_(self): 

""" 

Return print representation of this relative order. 

 

EXAMPLES:: 

 

sage: O = EquationOrder([x^2 + x + 1, x^3 - 2],'a,b') 

sage: O._repr_() 

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

""" 

#", ".join([str(b) for b in self.basis()]), 

return "%sRelative Order in %r" % ("Maximal " if self._is_maximal else "", self._K) 

 

def absolute_order(self, names='z'): 

""" 

Return underlying absolute order associated to this relative 

order. 

 

INPUT: 

 

- ``names`` -- string (default: 'z'); name of generator of absolute extension. 

 

.. note:: 

 

There *is* a default variable name, since this absolute 

order is frequently used for internal algorithms. 

 

EXAMPLES:: 

 

sage: R = EquationOrder([x^2 + 1, x^2 - 5], 'i,g'); R 

Relative Order in Number Field in i with defining polynomial x^2 + 1 over its base field 

sage: R.basis() 

[1, 6*i - g, -g*i + 2, 7*i - g] 

 

sage: S = R.absolute_order(); S 

Order in Number Field in z with defining polynomial x^4 - 8*x^2 + 36 

sage: S.basis() 

[1, 5/12*z^3 + 1/6*z, 1/2*z^2, 1/2*z^3] 

 

We compute a relative order in alpha0, alpha1, then make the 

number field that contains the absolute order be called 

gamma.:: 

 

sage: R = EquationOrder( [x^2 + 2, x^2 - 3], 'alpha'); R 

Relative Order in Number Field in alpha0 with defining polynomial x^2 + 2 over its base field 

sage: R.absolute_order('gamma') 

Order in Number Field in gamma with defining polynomial x^4 - 2*x^2 + 25 

sage: R.absolute_order('gamma').basis() 

[1/2*gamma^2 + 1/2, 7/10*gamma^3 + 1/10*gamma, gamma^2, gamma^3] 

""" 

if names == 'z' or names == ('z',): 

return self._absolute_order 

else: 

return self._absolute_order.change_names(names) 

 

def __reduce__(self): 

r""" 

Used for pickling. 

 

EXAMPLES:: 

 

sage: L.<a, b> = NumberField([x^2 + 1, x^2 - 5]) 

sage: O = L.maximal_order() 

sage: _ = O.basis() 

sage: O == loads(dumps(O)) 

True 

""" 

return (RelativeOrder, (self.number_field(), self.absolute_order(), self._is_maximal, False)) 

 

def basis(self): 

""" 

Return a basis for this order as `\ZZ`-module. 

 

EXAMPLES:: 

 

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

sage: O = K.order([a,b]) 

sage: O.basis() 

[1, -2*a + b, -b*a - 2, -5*a + 3*b] 

sage: z = O.1; z 

-2*a + b 

sage: z.absolute_minpoly() 

x^4 + 14*x^2 + 1 

""" 

try: 

return self.__basis 

except AttributeError: 

pass 

O = self._absolute_order 

K = O.number_field() 

from_K, _ = K.structure() 

self.__basis = [OrderElement_relative(self, from_K(a)) for a in O.basis()] 

return self.__basis 

 

def __add__(left, right): 

""" 

Add two relative orders or a relative order to an absolute 

order (which always results in an absolute order). 

 

EXAMPLES:: 

 

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

sage: O2 = K.order([2*a, b]); O2.absolute_discriminant() 

36864 

sage: O3 = K.order([3*a, 2*b]); O3.absolute_discriminant() 

2985984 

sage: R = O2 + O3; R 

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

sage: R.absolute_discriminant() 

9216 

sage: R.is_suborder(O2) 

False 

sage: O2.is_suborder(R) 

True 

sage: O3.is_suborder(R) 

True 

""" 

if isinstance(left, AbsoluteOrder): 

return left + right._absolute_order 

elif isinstance(right, AbsoluteOrder): 

return left._absolute_order + right 

elif isinstance(left, RelativeOrder) and isinstance(right, RelativeOrder): 

if left._K != right._K: 

raise TypeError("Number fields don't match.") 

return RelativeOrder(left._K, left._absolute_order + right._absolute_order, 

check=False) 

else: 

raise NotImplementedError 

 

def __and__(left, right): 

""" 

Intersect two relative orders or a relative and absolute order 

(which always results in an absolute order). 

 

EXAMPLES:: 

 

sage: L.<a, b> = NumberField([x^2 + 1, x^2 - 5]) 

sage: O1 = L.order([a, 2*b]) 

sage: O2 = L.order([2*a, b]) 

sage: O3 = O1 & O2; O3 

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

sage: O3.index_in(L.maximal_order()) 

32 

""" 

if isinstance(left, AbsoluteOrder): 

return left & right._absolute_order 

elif isinstance(right, AbsoluteOrder): 

return left._absolute_order & right 

elif isinstance(left, RelativeOrder) and isinstance(right, RelativeOrder): 

if left._K != right._K: 

raise TypeError("Number fields don't match.") 

return RelativeOrder(left._K, left._absolute_order & right._absolute_order, 

check=False) 

else: 

raise NotImplementedError 

 

def absolute_discriminant(self): 

""" 

Return the absolute discriminant of self, which is the discriminant 

of the absolute order associated to self. 

 

OUTPUT: 

 

an integer 

 

EXAMPLES:: 

 

sage: R = EquationOrder([x^2 + 1, x^3 + 2], 'a,b') 

sage: d = R.absolute_discriminant(); d 

-746496 

sage: d is R.absolute_discriminant() 

True 

sage: factor(d) 

-1 * 2^10 * 3^6 

""" 

return self.absolute_order().discriminant() 

 

def is_suborder(self, other): 

""" 

Returns true if self is a subset of the order other. 

 

EXAMPLES:: 

 

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

sage: R1 = K.order([a,b]) 

sage: R2 = K.order([2*a,b]) 

sage: R3 = K.order([a + b, b + 2*a]) 

sage: R1.is_suborder(R2) 

False 

sage: R2.is_suborder(R1) 

True 

sage: R3.is_suborder(R1) 

True 

sage: R1.is_suborder(R3) 

True 

sage: R1 == R3 

True 

""" 

return self.absolute_order().is_suborder(other.absolute_order()) 

 

def index_in(self, other): 

""" 

Return the index of self in other. This is a lattice index, 

so it is a rational number if self isn't contained in other. 

 

INPUT: 

 

- ``other`` -- another order with the same ambient absolute number field. 

 

OUTPUT: 

 

a rational number 

 

EXAMPLES:: 

 

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

sage: R1 = K.order([3*a, 2*b]) 

sage: R2 = K.order([a, 4*b]) 

sage: R1.index_in(R2) 

729/8 

sage: R2.index_in(R1) 

8/729 

""" 

if not isinstance(other, Order): 

raise TypeError("other must be an absolute order.") 

return self.absolute_order().index_in(other.absolute_order()) 

 

 

 

def each_is_integral(v): 

""" 

Return True if each element of the list ``v`` of elements of a number 

field is integral. 

 

EXAMPLES:: 

 

sage: W.<sqrt5> = NumberField(x^2 - 5) 

sage: from sage.rings.number_field.order import each_is_integral 

sage: each_is_integral([sqrt5, 2, (1+sqrt5)/2]) 

True 

sage: each_is_integral([sqrt5, (1+sqrt5)/3]) 

False 

""" 

for x in v: 

if not x.is_integral(): 

return False 

return True 

 

def absolute_order_from_ring_generators(gens, check_is_integral=True, 

check_rank=True, is_maximal=None, 

allow_subfield=False): 

""" 

INPUT: 

 

- ``gens`` -- list of integral elements of an absolute order. 

- ``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. 

- ``is_maximal`` -- bool (or None); set if maximality of the generated order is 

known 

- ``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.<a> = NumberField(x^4 - 5) 

sage: K.order(a) 

Order in Number Field in a with defining polynomial x^4 - 5 

 

We have to explicitly import this function, since typically it is called 

with ``K.order`` as above.:: 

 

sage: from sage.rings.number_field.order import absolute_order_from_ring_generators 

sage: absolute_order_from_ring_generators([a]) 

Order in Number Field in a with defining polynomial x^4 - 5 

sage: absolute_order_from_ring_generators([3*a, 2, 6*a+1]) 

Order in Number Field in a with defining polynomial x^4 - 5 

 

If one of the inputs is non-integral, it is an error.:: 

 

sage: absolute_order_from_ring_generators([a/2]) 

Traceback (most recent call last): 

... 

ValueError: each generator must be integral 

 

If the gens do not generate an order, i.e., generate a ring of full 

rank, then it is an error.:: 

 

sage: absolute_order_from_ring_generators([a^2]) 

Traceback (most recent call last): 

... 

ValueError: the rank of the span of gens is wrong 

 

Both checking for integrality and checking for full rank can be 

turned off in order to save time, though one can get nonsense as 

illustrated below.:: 

 

sage: absolute_order_from_ring_generators([a/2], check_is_integral=False) 

Order in Number Field in a with defining polynomial x^4 - 5 

sage: absolute_order_from_ring_generators([a^2], check_rank=False) 

Order in Number Field in a with defining polynomial x^4 - 5 

""" 

if check_is_integral and not each_is_integral(gens): 

raise ValueError("each generator must be integral") 

gens = Sequence(gens) 

K = gens.universe() 

n = [x.absolute_minpoly().degree() for x in gens] 

module_gens = monomials(gens, n) 

return absolute_order_from_module_generators(module_gens, 

check_integral=False, check_is_ring=False, 

check_rank=check_rank, is_maximal=is_maximal, 

allow_subfield = allow_subfield) 

 

 

def absolute_order_from_module_generators(gens, 

check_integral=True, check_rank=True, 

check_is_ring=True, is_maximal=None, 

allow_subfield = False): 

""" 

INPUT: 

 

- ``gens`` -- list of elements of an absolute number field that generates an 

order in that number field as a ZZ *module*. 

- ``check_integral`` -- check that each gen is integral 

- ``check_rank`` -- check that the gens span a module of the correct rank 

- ``check_is_ring`` -- check that the module is closed under multiplication 

(this is very expensive) 

- ``is_maximal`` -- bool (or None); set if maximality of the generated order is known 

 

OUTPUT: 

 

an absolute order 

 

EXAMPLES: 

 

We have to explicitly import the function, since it isn't meant 

for regular usage:: 

 

sage: from sage.rings.number_field.order import absolute_order_from_module_generators 

 

sage: K.<a> = NumberField(x^4 - 5) 

sage: O = K.maximal_order(); O 

Maximal Order in Number Field in a with defining polynomial x^4 - 5 

sage: O.basis() 

[1/2*a^2 + 1/2, 1/2*a^3 + 1/2*a, a^2, a^3] 

sage: O.module() 

Free module of degree 4 and rank 4 over Integer Ring 

Echelon basis matrix: 

[1/2 0 1/2 0] 

[ 0 1/2 0 1/2] 

[ 0 0 1 0] 

[ 0 0 0 1] 

sage: g = O.basis(); g 

[1/2*a^2 + 1/2, 1/2*a^3 + 1/2*a, a^2, a^3] 

sage: absolute_order_from_module_generators(g) 

Order in Number Field in a with defining polynomial x^4 - 5 

 

We illustrate each check flag -- the output is the same but in case 

the function would run ever so slightly faster:: 

 

sage: absolute_order_from_module_generators(g, check_is_ring=False) 

Order in Number Field in a with defining polynomial x^4 - 5 

sage: absolute_order_from_module_generators(g, check_rank=False) 

Order in Number Field in a with defining polynomial x^4 - 5 

sage: absolute_order_from_module_generators(g, check_integral=False) 

Order in Number Field in a with defining polynomial x^4 - 5 

 

Next we illustrate constructing "fake" orders to illustrate turning 

off various check flags:: 

 

sage: k.<i> = NumberField(x^2 + 1) 

sage: R = absolute_order_from_module_generators([2, 2*i], check_is_ring=False); R 

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

sage: R.basis() 

[2, 2*i] 

sage: R = absolute_order_from_module_generators([k(1)], check_rank=False); R 

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

sage: R.basis() 

[1] 

 

If the order contains a non-integral element, even if we don't check 

that, we'll find that the rank is wrong or that the order isn't closed 

under multiplication:: 

 

sage: absolute_order_from_module_generators([1/2, i], check_integral=False) 

Traceback (most recent call last): 

... 

ValueError: the module span of the gens is not closed under multiplication. 

sage: R = absolute_order_from_module_generators([1/2, i], check_is_ring=False, check_integral=False); R 

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

sage: R.basis() 

[1/2, i] 

 

We turn off all check flags and make a really messed up order:: 

 

sage: R = absolute_order_from_module_generators([1/2, i], check_is_ring=False, check_integral=False, check_rank=False); R 

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

sage: R.basis() 

[1/2, i] 

 

An order that lives in a subfield:: 

 

sage: F.<alpha> = NumberField(x**4+3) 

sage: F.order([alpha**2], allow_subfield=True) 

Order in Number Field in alpha with defining polynomial x^4 + 3 

""" 

if len(gens) == 0: 

raise ValueError("gens must span an order over ZZ") 

gens = Sequence(gens) 

if check_integral and not each_is_integral(gens): 

raise ValueError("each generator must be integral") 

 

K = gens.universe() 

if is_NumberFieldOrder(K): 

K = K.number_field() 

V, from_V, to_V = K.vector_space() 

mod_gens = [to_V(x) for x in gens] 

ambient = ZZ**V.dimension() 

W = ambient.span(mod_gens) 

 

if allow_subfield: 

if W.rank() < K.degree(): 

# We have to make the order in a smaller field. 

# We do this by choosing a random element of W, 

# moving it back to K, and checking that it defines 

# a field of degree equal to the degree of W. 

# Then we move everything into that field, where 

# W does define an order. 

while True: 

z = V.random_element() 

alpha = from_V(z) 

if alpha.minpoly().degree() == W.rank(): 

break 

# Now alpha generates a subfield there W is an order 

# (with the right rank). 

# We move each element of W to this subfield. 

c = alpha.coordinates_in_terms_of_powers() 

 

elif check_rank: 

if W.rank() != K.degree(): 

raise ValueError("the rank of the span of gens is wrong") 

 

if check_is_ring: 

# Is there a faster way? 

alg = [to_V(x) for x in monomials(gens, [f.absolute_minpoly().degree() for f in gens])] 

if ambient.span(alg) != W: 

raise ValueError("the module span of the gens is not closed under multiplication.") 

 

return AbsoluteOrder(K, W, check=False, is_maximal=is_maximal) # we have already checked everything 

 

 

 

 

 

def relative_order_from_ring_generators(gens, 

check_is_integral=True, 

check_rank=True, 

is_maximal = None, 

allow_subfield=False): 

""" 

INPUT: 

 

- ``gens`` -- list of integral elements of an absolute order. 

- ``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. 

- ``is_maximal`` -- bool (or None); set if maximality of the generated order is 

known 

 

EXAMPLES: 

 

We have to explicitly import this function, since it isn't meant 

for regular usage:: 

 

sage: from sage.rings.number_field.order import relative_order_from_ring_generators 

sage: K.<i, a> = NumberField([x^2 + 1, x^2 - 17]) 

sage: R = K.base_field().maximal_order() 

sage: S = relative_order_from_ring_generators([i,a]); S 

Relative Order in Number Field in i with defining polynomial x^2 + 1 over its base field 

 

Basis for the relative order, which is obtained by computing the algebra generated 

by i and a:: 

 

sage: S.basis() 

[1, 7*i - 2*a, -a*i + 8, 25*i - 7*a] 

""" 

if check_is_integral and not each_is_integral(gens): 

raise ValueError("each generator must be integral") 

gens = Sequence(gens) 

 

# The top number field that contains the order. 

K = gens.universe() 

 

# The absolute version of that field. 

Kabs = K.absolute_field('z') 

from_Kabs, to_Kabs = Kabs.structure() 

 

module_gens = [to_Kabs(a) for a in gens] 

n = [a.absolute_minpoly().degree() for a in gens] 

absolute_order_module_gens = monomials(module_gens, n) 

 

abs_order = absolute_order_from_module_generators(absolute_order_module_gens, 

check_integral=False, check_is_ring=False, 

check_rank=check_rank) 

 

return RelativeOrder(K, abs_order, check=False, is_maximal=is_maximal) 

 

 

def GaussianIntegers(names="I"): 

""" 

Return the ring of Gaussian integers, that is all complex numbers 

of the form `a + b I` with `a` and `b` integers and `I = \sqrt{-1}`. 

 

EXAMPLES:: 

 

sage: ZZI.<I> = GaussianIntegers() 

sage: ZZI 

Gaussian Integers in Number Field in I with defining polynomial x^2 + 1 

sage: factor(3 + I) 

(-I) * (I + 1) * (2*I + 1) 

sage: CC(I) 

1.00000000000000*I 

sage: I.minpoly() 

x^2 + 1 

sage: GaussianIntegers().basis() 

[1, I] 

""" 

from sage.rings.all import CDF, NumberField 

f = ZZ['x']([1,0,1]) 

nf = NumberField(f, names, embedding=CDF(0, 1)) 

return nf.ring_of_integers() 

 

 

def EisensteinIntegers(names="omega"): 

""" 

Return the ring of Eisenstein integers, that is all complex numbers 

of the form `a + b \omega` with `a` and `b` integers and 

`omega = (-1 + \sqrt{-3})/2`. 

 

EXAMPLES:: 

 

sage: R.<omega> = EisensteinIntegers() 

sage: R 

Eisenstein Integers in Number Field in omega with defining polynomial x^2 + x + 1 

sage: factor(3 + omega) 

(omega) * (-3*omega - 2) 

sage: CC(omega) 

-0.500000000000000 + 0.866025403784439*I 

sage: omega.minpoly() 

x^2 + x + 1 

sage: EisensteinIntegers().basis() 

[1, omega] 

""" 

from sage.rings.all import CDF, NumberField 

f = ZZ['x']([1,1,1]) 

nf = NumberField(f, names, embedding=CDF(-0.5, 0.8660254037844386)) 

return nf.ring_of_integers()