Ideals of number fields¶

AUTHORS:

Steven Sivek (2005-05-16): initial version
William Stein (2007-09-06): vastly improved the doctesting
William Stein and John Cremona (2007-01-28): new class NumberFieldFractionalIdeal now used for all except the 0 ideal
Radoslav Kirov and Alyson Deines (2010-06-22): prime_to_S_part, is_S_unit, is_S_integral

class sage.rings.number_field.number_field_ideal.LiftMap(OK, M_OK_map, Q, I)[source]¶

Bases: object

Class to hold data needed by lifting maps from residue fields to number field orders.

class sage.rings.number_field.number_field_ideal.NumberFieldFractionalIdeal(field, gens, coerce=True)[source]¶

Bases: MultiplicativeGroupElement, NumberFieldIdeal, Ideal_fractional

A fractional ideal in a number field.

EXAMPLES:

Sage

sage: x = polygen(ZZ)
sage: R.<x> = PolynomialRing(QQ)
sage: K.<a> = NumberField(x^3 - 2)
sage: I = K.ideal(2/(5+a))
sage: J = I^2
sage: Jinv = I^(-2)
sage: J*Jinv
Fractional ideal (1)

Python

>>> from sage.all import *
>>> x = polygen(ZZ)
>>> R = PolynomialRing(QQ, names=('x',)); (x,) = R._first_ngens(1)
>>> K = NumberField(x**Integer(3) - Integer(2), names=('a',)); (a,) = K._first_ngens(1)
>>> I = K.ideal(Integer(2)/(Integer(5)+a))
>>> J = I**Integer(2)
>>> Jinv = I**(-Integer(2))
>>> J*Jinv
Fractional ideal (1)

Sage Live

x = polygen(ZZ)
R.<x> = PolynomialRing(QQ)
K.<a> = NumberField(x^3 - 2)
I = K.ideal(2/(5+a))
J = I^2
Jinv = I^(-2)
J*Jinv

denominator()[source]¶

Return the denominator ideal of this fractional ideal. Each fractional ideal has a unique expression as \(N/D\) where \(N\), \(D\) are coprime integral ideals; the denominator is \(D\).

EXAMPLES:

Sage

sage: x = polygen(ZZ)
sage: K.<i> = NumberField(x^2 + 1)
sage: I = K.ideal((3+4*i)/5); I
Fractional ideal (4/5*i + 3/5)
sage: I.denominator()
Fractional ideal (2*i + 1)
sage: I.numerator()
Fractional ideal (-i - 2)
sage: I.numerator().is_integral() and I.denominator().is_integral()
True
sage: I.numerator() + I.denominator() == K.unit_ideal()
True
sage: I.numerator()/I.denominator() == I
True

Python

>>> from sage.all import *
>>> x = polygen(ZZ)
>>> K = NumberField(x**Integer(2) + Integer(1), names=('i',)); (i,) = K._first_ngens(1)
>>> I = K.ideal((Integer(3)+Integer(4)*i)/Integer(5)); I
Fractional ideal (4/5*i + 3/5)
>>> I.denominator()
Fractional ideal (2*i + 1)
>>> I.numerator()
Fractional ideal (-i - 2)
>>> I.numerator().is_integral() and I.denominator().is_integral()
True
>>> I.numerator() + I.denominator() == K.unit_ideal()
True
>>> I.numerator()/I.denominator() == I
True

Sage Live

x = polygen(ZZ)
K.<i> = NumberField(x^2 + 1)
I = K.ideal((3+4*i)/5); I
I.denominator()
I.numerator()
I.numerator().is_integral() and I.denominator().is_integral()
I.numerator() + I.denominator() == K.unit_ideal()
I.numerator()/I.denominator() == I

divides(other)[source]¶

Return True if this ideal divides other and False otherwise.

EXAMPLES:

Sage

sage: K.<a> = CyclotomicField(11); K
Cyclotomic Field of order 11 and degree 10
sage: I = K.factor(31)[0][0]; I
Fractional ideal (31, a^5 + 10*a^4 - a^3 + a^2 + 9*a - 1)
sage: I.divides(I)
True
sage: I.divides(31)
True
sage: I.divides(29)
False

Python

>>> from sage.all import *
>>> K = CyclotomicField(Integer(11), names=('a',)); (a,) = K._first_ngens(1); K
Cyclotomic Field of order 11 and degree 10
>>> I = K.factor(Integer(31))[Integer(0)][Integer(0)]; I
Fractional ideal (31, a^5 + 10*a^4 - a^3 + a^2 + 9*a - 1)
>>> I.divides(I)
True
>>> I.divides(Integer(31))
True
>>> I.divides(Integer(29))
False

Sage Live

K.<a> = CyclotomicField(11); K
I = K.factor(31)[0][0]; I
I.divides(I)
I.divides(31)
I.divides(29)

element_1_mod(other)[source]¶

Return an element \(r\) in this ideal such that \(1-r\) is in other.

An error is raised if either ideal is not integral of if they are not coprime.

INPUT:

other – another ideal of the same field, or generators of an ideal

OUTPUT: an element \(r\) of the ideal self such that \(1-r\) is in the ideal other

AUTHOR: Maite Aranes (modified to use PARI’s pari:idealaddtoone by Francis Clarke)

EXAMPLES:

Sage

sage: x = polygen(ZZ)
sage: K.<a> = NumberField(x^3 - 2)
sage: A = K.ideal(a + 1); A; A.norm()
Fractional ideal (a + 1)
3
sage: B = K.ideal(a^2 - 4*a + 2); B; B.norm()
Fractional ideal (a^2 - 4*a + 2)
68
sage: r = A.element_1_mod(B); r
-33
sage: r in A
True
sage: 1 - r in B
True

Python

>>> from sage.all import *
>>> x = polygen(ZZ)
>>> K = NumberField(x**Integer(3) - Integer(2), names=('a',)); (a,) = K._first_ngens(1)
>>> A = K.ideal(a + Integer(1)); A; A.norm()
Fractional ideal (a + 1)
3
>>> B = K.ideal(a**Integer(2) - Integer(4)*a + Integer(2)); B; B.norm()
Fractional ideal (a^2 - 4*a + 2)
68
>>> r = A.element_1_mod(B); r
-33
>>> r in A
True
>>> Integer(1) - r in B
True

Sage Live

x = polygen(ZZ)
K.<a> = NumberField(x^3 - 2)
A = K.ideal(a + 1); A; A.norm()
B = K.ideal(a^2 - 4*a + 2); B; B.norm()
r = A.element_1_mod(B); r
r in A
1 - r in B

euler_phi()[source]¶

Return the Euler \(\varphi\)-function of this integral ideal.

This is the order of the multiplicative group of the quotient modulo the ideal.

An error is raised if the ideal is not integral.

EXAMPLES:

Sage

sage: x = polygen(ZZ)
sage: K.<i> = NumberField(x^2 + 1)
sage: I = K.ideal(2 + i)
sage: [r for r in I.residues() if I.is_coprime(r)]
[-2*i, -i, i, 2*i]
sage: I.euler_phi()
4
sage: J = I^3
sage: J.euler_phi()
100
sage: len([r for r in J.residues() if J.is_coprime(r)])
100
sage: J = K.ideal(3 - 2*i)
sage: I.is_coprime(J)
True
sage: I.euler_phi()*J.euler_phi() == (I*J).euler_phi()
True
sage: L.<b> = K.extension(x^2 - 7)
sage: L.ideal(3).euler_phi()
64

Python

>>> from sage.all import *
>>> x = polygen(ZZ)
>>> K = NumberField(x**Integer(2) + Integer(1), names=('i',)); (i,) = K._first_ngens(1)
>>> I = K.ideal(Integer(2) + i)
>>> [r for r in I.residues() if I.is_coprime(r)]
[-2*i, -i, i, 2*i]
>>> I.euler_phi()
4
>>> J = I**Integer(3)
>>> J.euler_phi()
100
>>> len([r for r in J.residues() if J.is_coprime(r)])
100
>>> J = K.ideal(Integer(3) - Integer(2)*i)
>>> I.is_coprime(J)
True
>>> I.euler_phi()*J.euler_phi() == (I*J).euler_phi()
True
>>> L = K.extension(x**Integer(2) - Integer(7), names=('b',)); (b,) = L._first_ngens(1)
>>> L.ideal(Integer(3)).euler_phi()
64

Sage Live

x = polygen(ZZ)
K.<i> = NumberField(x^2 + 1)
I = K.ideal(2 + i)
[r for r in I.residues() if I.is_coprime(r)]
I.euler_phi()
J = I^3
J.euler_phi()
len([r for r in J.residues() if J.is_coprime(r)])
J = K.ideal(3 - 2*i)
I.is_coprime(J)
I.euler_phi()*J.euler_phi() == (I*J).euler_phi()
L.<b> = K.extension(x^2 - 7)
L.ideal(3).euler_phi()

factor()[source]¶

Factorization of this ideal in terms of prime ideals.

EXAMPLES:

Sage

sage: x = polygen(ZZ)
sage: K.<a> = NumberField(x^4 + 23); K
Number Field in a with defining polynomial x^4 + 23
sage: I = K.ideal(19); I
Fractional ideal (19)
sage: F = I.factor(); F
(Fractional ideal (19, 1/2*a^2 + a - 17/2))
 * (Fractional ideal (19, 1/2*a^2 - a - 17/2))
sage: type(F)
<class 'sage.structure.factorization.Factorization'>
sage: list(F)
[(Fractional ideal (19, 1/2*a^2 + a - 17/2), 1),
 (Fractional ideal (19, 1/2*a^2 - a - 17/2), 1)]
sage: F.prod()
Fractional ideal (19)

Python

>>> from sage.all import *
>>> x = polygen(ZZ)
>>> K = NumberField(x**Integer(4) + Integer(23), names=('a',)); (a,) = K._first_ngens(1); K
Number Field in a with defining polynomial x^4 + 23
>>> I = K.ideal(Integer(19)); I
Fractional ideal (19)
>>> F = I.factor(); F
(Fractional ideal (19, 1/2*a^2 + a - 17/2))
 * (Fractional ideal (19, 1/2*a^2 - a - 17/2))
>>> type(F)
<class 'sage.structure.factorization.Factorization'>
>>> list(F)
[(Fractional ideal (19, 1/2*a^2 + a - 17/2), 1),
 (Fractional ideal (19, 1/2*a^2 - a - 17/2), 1)]
>>> F.prod()
Fractional ideal (19)

Sage Live

x = polygen(ZZ)
K.<a> = NumberField(x^4 + 23); K
I = K.ideal(19); I
F = I.factor(); F
type(F)
list(F)
F.prod()

idealcoprime(J)[source]¶

Return \(l\) such that l*self is coprime to \(J\).

INPUT:

J – another integral ideal of the same field as self, which must also be integral

OUTPUT: an element \(l\) such that l*self is coprime to the ideal \(J\)

Todo

Extend the implementation to non-integral ideals.

EXAMPLES:

Sage

sage: x = polygen(ZZ)
sage: k.<a> = NumberField(x^2 + 23)
sage: A = k.ideal(a + 1)
sage: B = k.ideal(3)
sage: A.is_coprime(B)
False
sage: lam = A.idealcoprime(B)
sage: lam  # representation depends, not tested
-1/6*a + 1/6
sage: (lam*A).is_coprime(B)
True

Python

>>> from sage.all import *
>>> x = polygen(ZZ)
>>> k = NumberField(x**Integer(2) + Integer(23), names=('a',)); (a,) = k._first_ngens(1)
>>> A = k.ideal(a + Integer(1))
>>> B = k.ideal(Integer(3))
>>> A.is_coprime(B)
False
>>> lam = A.idealcoprime(B)
>>> lam  # representation depends, not tested
-1/6*a + 1/6
>>> (lam*A).is_coprime(B)
True

Sage Live

x = polygen(ZZ)
k.<a> = NumberField(x^2 + 23)
A = k.ideal(a + 1)
B = k.ideal(3)
A.is_coprime(B)
lam = A.idealcoprime(B)
lam  # representation depends, not tested
(lam*A).is_coprime(B)

ALGORITHM: Uses Pari function pari:idealcoprime.

ideallog(x, gens=None, check=True)[source]¶

Return the discrete logarithm of \(x\) with respect to the generators given in the bid structure of the ideal self, or with respect to the generators gens if these are given.

INPUT:

x – a nonzero element of the number field of self, which must have valuation equal to 0 at all prime ideals in the support of the ideal self
gens – list of elements of the number field which generate \((R / I)^*\), where \(R\) is the ring of integers of the field and \(I\) is this ideal, or None. If None, use the generators calculated by idealstar().
check – if True, do a consistency check on the results. Ignored if gens is None.

OUTPUT:

a list of nonnegative integers \((x_i)\) such that \(x = \prod_i g_i^{x_i}\) in \((R/I)^*\), where \(x_i\) are the generators, and the list \((x_i)\) is lexicographically minimal with respect to this requirement. If the \(x_i\) generate independent cyclic factors of order \(d_i\), as is the case for the default generators calculated by idealstar(), this just means that \(0 \le x_i < d_i\).

A ValueError will be raised if the elements specified in gens do not in fact generate the unit group (even if the element \(x\) is in the subgroup they generate).

EXAMPLES:

Sage

sage: x = polygen(ZZ)
sage: k.<a> = NumberField(x^3 - 11)
sage: A = k.ideal(5)
sage: G = A.idealstar(2)
sage: l = A.ideallog(a^2 + 3)
sage: r = G(l).value()
sage: (a^2 + 3) - r in A
True
sage: A.small_residue(r) # random
a^2 - 2

Python

>>> from sage.all import *
>>> x = polygen(ZZ)
>>> k = NumberField(x**Integer(3) - Integer(11), names=('a',)); (a,) = k._first_ngens(1)
>>> A = k.ideal(Integer(5))
>>> G = A.idealstar(Integer(2))
>>> l = A.ideallog(a**Integer(2) + Integer(3))
>>> r = G(l).value()
>>> (a**Integer(2) + Integer(3)) - r in A
True
>>> A.small_residue(r) # random
a^2 - 2

Sage Live

x = polygen(ZZ)
k.<a> = NumberField(x^3 - 11)
A = k.ideal(5)
G = A.idealstar(2)
l = A.ideallog(a^2 + 3)
r = G(l).value()
(a^2 + 3) - r in A
A.small_residue(r) # random

Examples with custom generators:

Sage

sage: K.<a> = NumberField(x^2 - 7)
sage: I = K.ideal(17)
sage: I.ideallog(a + 7, [1 + a, 2])
[10, 3]
sage: I.ideallog(a + 7, [2, 1 + a])
[0, 118]

sage: L.<b> = NumberField(x^4 - x^3 - 7*x^2 + 3*x + 2)
sage: J = L.ideal(-b^3 - b^2 - 2)
sage: u = -14*b^3 + 21*b^2 + b - 1
sage: v = 4*b^2 + 2*b - 1
sage: J.ideallog(5 + 2*b, [u, v], check=True)
[4, 13]

Python

>>> from sage.all import *
>>> K = NumberField(x**Integer(2) - Integer(7), names=('a',)); (a,) = K._first_ngens(1)
>>> I = K.ideal(Integer(17))
>>> I.ideallog(a + Integer(7), [Integer(1) + a, Integer(2)])
[10, 3]
>>> I.ideallog(a + Integer(7), [Integer(2), Integer(1) + a])
[0, 118]

>>> L = NumberField(x**Integer(4) - x**Integer(3) - Integer(7)*x**Integer(2) + Integer(3)*x + Integer(2), names=('b',)); (b,) = L._first_ngens(1)
>>> J = L.ideal(-b**Integer(3) - b**Integer(2) - Integer(2))
>>> u = -Integer(14)*b**Integer(3) + Integer(21)*b**Integer(2) + b - Integer(1)
>>> v = Integer(4)*b**Integer(2) + Integer(2)*b - Integer(1)
>>> J.ideallog(Integer(5) + Integer(2)*b, [u, v], check=True)
[4, 13]

Sage Live

K.<a> = NumberField(x^2 - 7)
I = K.ideal(17)
I.ideallog(a + 7, [1 + a, 2])
I.ideallog(a + 7, [2, 1 + a])
L.<b> = NumberField(x^4 - x^3 - 7*x^2 + 3*x + 2)
J = L.ideal(-b^3 - b^2 - 2)
u = -14*b^3 + 21*b^2 + b - 1
v = 4*b^2 + 2*b - 1
J.ideallog(5 + 2*b, [u, v], check=True)

A non-example:

Sage

sage: I.ideallog(a + 7, [2])
Traceback (most recent call last):
...
ValueError: Given elements do not generate unit group --
they generate a subgroup of index 36

Python

>>> from sage.all import *
>>> I.ideallog(a + Integer(7), [Integer(2)])
Traceback (most recent call last):
...
ValueError: Given elements do not generate unit group --
they generate a subgroup of index 36

Sage Live

I.ideallog(a + 7, [2])

ALGORITHM: Uses PARI function pari:ideallog, and (if gens is not None) a Hermite normal form calculation to express the result in terms of the generators gens.

idealstar(flag=1)[source]¶

Return the finite abelian group \((O_K/I)^*\), where \(I\) is the ideal self of the number field \(K\), and \(O_K\) is the ring of integers of \(K\).

INPUT:

flag – integer (default: 1); when flag==2, it also computes the generators of the group \((O_K/I)^*\), which takes more time. By default flag is 1 (no generators are computed). In both cases the special PARI structure bid is computed as well. If flag is 0 (deprecated) it computes only the group structure of \((O_K/I)^*\) (with generators) and not the special bid structure.

OUTPUT:

The finite abelian group \((O_K/I)^*\).

Note

Uses the PARI function pari:idealstar. The PARI function outputs a special bid structure which is stored in the internal field _bid of the ideal (when flag = 1,2). The special structure bid is used in the PARI function pari:ideallog to compute discrete logarithms.

EXAMPLES:

Sage

sage: x = polygen(ZZ)
sage: k.<a> = NumberField(x^3 - 11)
sage: A = k.ideal(5)
sage: G = A.idealstar(); G
Multiplicative Abelian group isomorphic to C24 x C4
sage: G.gens()
(f0, f1)

sage: G = A.idealstar(2)
sage: G.gens()
(f0, f1)
sage: G.gens_values()   # random output
(2*a^2 - 1, 2*a^2 + 2*a - 2)
sage: all(G.gen(i).value() in k for i in range(G.ngens()))
True

Python

>>> from sage.all import *
>>> x = polygen(ZZ)
>>> k = NumberField(x**Integer(3) - Integer(11), names=('a',)); (a,) = k._first_ngens(1)
>>> A = k.ideal(Integer(5))
>>> G = A.idealstar(); G
Multiplicative Abelian group isomorphic to C24 x C4
>>> G.gens()
(f0, f1)

>>> G = A.idealstar(Integer(2))
>>> G.gens()
(f0, f1)
>>> G.gens_values()   # random output
(2*a^2 - 1, 2*a^2 + 2*a - 2)
>>> all(G.gen(i).value() in k for i in range(G.ngens()))
True

Sage Live

x = polygen(ZZ)
k.<a> = NumberField(x^3 - 11)
A = k.ideal(5)
G = A.idealstar(); G
G.gens()
G = A.idealstar(2)
G.gens()
G.gens_values()   # random output
all(G.gen(i).value() in k for i in range(G.ngens()))

ALGORITHM: Uses Pari function pari:idealstar

invertible_residues(reduce=True)[source]¶

Return an iterator through a list of invertible residues modulo this integral ideal.

An error is raised if this fractional ideal is not integral.

INPUT:

reduce – boolean; if True (default), use small_residue to get small representatives of the residues

OUTPUT:

An iterator through a list of invertible residues modulo this ideal \(I\), i.e. a list of elements in the ring of integers \(R\) representing the elements of \((R/I)^*\).

ALGORITHM: Use pari:idealstar to find the group structure and generators of the multiplicative group modulo the ideal.

EXAMPLES:

Sage

sage: x = polygen(ZZ)
sage: K.<i> = NumberField(x^2 + 1)
sage: ires = K.ideal(2).invertible_residues(); ires
xmrange_iter([[0, 1]], <function ...<lambda> at 0x...>)
sage: list(ires)
[1, -i]
sage: list(K.ideal(2 + i).invertible_residues())
[1, 2, 4, 3]
sage: list(K.ideal(i).residues())
[0]
sage: list(K.ideal(i).invertible_residues())
[1]
sage: I = K.ideal(3 + 6*i)
sage: units = I.invertible_residues()
sage: len(list(units)) == I.euler_phi()
True

sage: K.<a> = NumberField(x^3 - 10)
sage: I = K.ideal(a - 1)
sage: len(list(I.invertible_residues())) == I.euler_phi()
True

sage: K.<z> = CyclotomicField(10)
sage: len(list(K.primes_above(3)[0].invertible_residues()))
80

Python

>>> from sage.all import *
>>> x = polygen(ZZ)
>>> K = NumberField(x**Integer(2) + Integer(1), names=('i',)); (i,) = K._first_ngens(1)
>>> ires = K.ideal(Integer(2)).invertible_residues(); ires
xmrange_iter([[0, 1]], <function ...<lambda> at 0x...>)
>>> list(ires)
[1, -i]
>>> list(K.ideal(Integer(2) + i).invertible_residues())
[1, 2, 4, 3]
>>> list(K.ideal(i).residues())
[0]
>>> list(K.ideal(i).invertible_residues())
[1]
>>> I = K.ideal(Integer(3) + Integer(6)*i)
>>> units = I.invertible_residues()
>>> len(list(units)) == I.euler_phi()
True

>>> K = NumberField(x**Integer(3) - Integer(10), names=('a',)); (a,) = K._first_ngens(1)
>>> I = K.ideal(a - Integer(1))
>>> len(list(I.invertible_residues())) == I.euler_phi()
True

>>> K = CyclotomicField(Integer(10), names=('z',)); (z,) = K._first_ngens(1)
>>> len(list(K.primes_above(Integer(3))[Integer(0)].invertible_residues()))
80

Sage Live

x = polygen(ZZ)
K.<i> = NumberField(x^2 + 1)
ires = K.ideal(2).invertible_residues(); ires
list(ires)
list(K.ideal(2 + i).invertible_residues())
list(K.ideal(i).residues())
list(K.ideal(i).invertible_residues())
I = K.ideal(3 + 6*i)
units = I.invertible_residues()
len(list(units)) == I.euler_phi()
K.<a> = NumberField(x^3 - 10)
I = K.ideal(a - 1)
len(list(I.invertible_residues())) == I.euler_phi()
K.<z> = CyclotomicField(10)
len(list(K.primes_above(3)[0].invertible_residues()))

AUTHOR: John Cremona

invertible_residues_mod(subgp_gens=[], reduce=True)[source]¶

Return a iterator through a list of representatives for the invertible residues modulo this integral ideal, modulo the subgroup generated by the elements in the list subgp_gens.

INPUT:

subgp_gens – either None or a list of elements of the number field of self. These need not be integral, but should be coprime to the ideal self. If the list is empty or None, the function returns an iterator through a list of representatives for the invertible residues modulo the integral ideal self.
reduce – boolean; if True (default), use small_residues to get small representatives of the residues

Note

See also invertible_residues() for a simpler version without the subgroup.

OUTPUT:

An iterator through a list of representatives for the invertible residues modulo self and modulo the group generated by subgp_gens, i.e. a list of elements in the ring of integers \(R\) representing the elements of \((R/I)^*/U\), where \(I\) is this ideal and \(U\) is the subgroup of \((R/I)^*\) generated by subgp_gens.

EXAMPLES:

Sage

sage: x = polygen(ZZ)
sage: k.<a> = NumberField(x^2 + 23)
sage: I = k.ideal(a)
sage: list(I.invertible_residues_mod([-1]))
[1, 5, 2, 10, 4, 20, 8, 17, 16, 11, 9]
sage: list(I.invertible_residues_mod([1/2]))
[1, 5]
sage: list(I.invertible_residues_mod([23]))
Traceback (most recent call last):
...
TypeError: the element must be invertible mod the ideal

Python

>>> from sage.all import *
>>> x = polygen(ZZ)
>>> k = NumberField(x**Integer(2) + Integer(23), names=('a',)); (a,) = k._first_ngens(1)
>>> I = k.ideal(a)
>>> list(I.invertible_residues_mod([-Integer(1)]))
[1, 5, 2, 10, 4, 20, 8, 17, 16, 11, 9]
>>> list(I.invertible_residues_mod([Integer(1)/Integer(2)]))
[1, 5]
>>> list(I.invertible_residues_mod([Integer(23)]))
Traceback (most recent call last):
...
TypeError: the element must be invertible mod the ideal

Sage Live

x = polygen(ZZ)
k.<a> = NumberField(x^2 + 23)
I = k.ideal(a)
list(I.invertible_residues_mod([-1]))
list(I.invertible_residues_mod([1/2]))
list(I.invertible_residues_mod([23]))

Sage

sage: K.<a> = NumberField(x^3 - 10)
sage: I = K.ideal(a - 1)
sage: len(list(I.invertible_residues_mod([]))) == I.euler_phi()
True

sage: I = K.ideal(1)
sage: list(I.invertible_residues_mod([]))
[1]

Python

>>> from sage.all import *
>>> K = NumberField(x**Integer(3) - Integer(10), names=('a',)); (a,) = K._first_ngens(1)
>>> I = K.ideal(a - Integer(1))
>>> len(list(I.invertible_residues_mod([]))) == I.euler_phi()
True

>>> I = K.ideal(Integer(1))
>>> list(I.invertible_residues_mod([]))
[1]

Sage Live

K.<a> = NumberField(x^3 - 10)
I = K.ideal(a - 1)
len(list(I.invertible_residues_mod([]))) == I.euler_phi()
I = K.ideal(1)
list(I.invertible_residues_mod([]))

Python

>>> from sage.all import *
>>> K = NumberField(x**Integer(3) - Integer(10), names=('a',)); (a,) = K._first_ngens(1)
>>> I = K.ideal(a - Integer(1))
>>> len(list(I.invertible_residues_mod([]))) == I.euler_phi()
True

>>> I = K.ideal(Integer(1))
>>> list(I.invertible_residues_mod([]))
[1]

Sage Live

K.<a> = NumberField(x^3 - 10)
I = K.ideal(a - 1)
len(list(I.invertible_residues_mod([]))) == I.euler_phi()
I = K.ideal(1)
list(I.invertible_residues_mod([]))

Sage

sage: K.<z> = CyclotomicField(10)
sage: len(list(K.primes_above(3)[0].invertible_residues_mod([])))
80

Python

>>> from sage.all import *
>>> K = CyclotomicField(Integer(10), names=('z',)); (z,) = K._first_ngens(1)
>>> len(list(K.primes_above(Integer(3))[Integer(0)].invertible_residues_mod([])))
80

Sage Live

K.<z> = CyclotomicField(10)
len(list(K.primes_above(3)[0].invertible_residues_mod([])))

Python

>>> from sage.all import *
>>> K = CyclotomicField(Integer(10), names=('z',)); (z,) = K._first_ngens(1)
>>> len(list(K.primes_above(Integer(3))[Integer(0)].invertible_residues_mod([])))
80

Sage Live

K.<z> = CyclotomicField(10)
len(list(K.primes_above(3)[0].invertible_residues_mod([])))

AUTHOR: Maite Aranes.

is_S_integral(S)[source]¶

Return True if this fractional ideal is integral with respect to the list of primes \(S\).

INPUT:

S – list of prime ideals (not checked if they are indeed prime)

Note

This function assumes that \(S\) is a list of prime ideals, but does not check this. This function will fail if \(S\) is not a list of prime ideals.

OUTPUT:

True, if the ideal is \(S\)-integral: that is, if the valuations of the ideal at all primes not in \(S\) are nonnegative. False, otherwise.

EXAMPLES:

Sage

sage: x = polygen(ZZ)
sage: K.<a> = NumberField(x^2 + 23)
sage: I = K.ideal(1/2)
sage: P = K.ideal(2, 1/2*a - 1/2)
sage: I.is_S_integral([P])
False

sage: J = K.ideal(1/5)
sage: J.is_S_integral([K.ideal(5)])
True

Python

>>> from sage.all import *
>>> x = polygen(ZZ)
>>> K = NumberField(x**Integer(2) + Integer(23), names=('a',)); (a,) = K._first_ngens(1)
>>> I = K.ideal(Integer(1)/Integer(2))
>>> P = K.ideal(Integer(2), Integer(1)/Integer(2)*a - Integer(1)/Integer(2))
>>> I.is_S_integral([P])
False

>>> J = K.ideal(Integer(1)/Integer(5))
>>> J.is_S_integral([K.ideal(Integer(5))])
True

Sage Live

x = polygen(ZZ)
K.<a> = NumberField(x^2 + 23)
I = K.ideal(1/2)
P = K.ideal(2, 1/2*a - 1/2)
I.is_S_integral([P])
J = K.ideal(1/5)
J.is_S_integral([K.ideal(5)])

is_S_unit(S)[source]¶

Return True if this fractional ideal is a unit with respect to the list of primes \(S\).

INPUT:

S – list of prime ideals (not checked if they are indeed prime)

Note

This function assumes that \(S\) is a list of prime ideals, but does not check this. This function will fail if \(S\) is not a list of prime ideals.

OUTPUT:

True, if the ideal is an \(S\)-unit: that is, if the valuations of the ideal at all primes not in \(S\) are zero. False, otherwise.

EXAMPLES:

Sage

sage: x = polygen(ZZ)
sage: K.<a> = NumberField(x^2 + 23)
sage: I = K.ideal(2)
sage: P = I.factor()[0][0]
sage: I.is_S_unit([P])
False

Python

>>> from sage.all import *
>>> x = polygen(ZZ)
>>> K = NumberField(x**Integer(2) + Integer(23), names=('a',)); (a,) = K._first_ngens(1)
>>> I = K.ideal(Integer(2))
>>> P = I.factor()[Integer(0)][Integer(0)]
>>> I.is_S_unit([P])
False

Sage Live

x = polygen(ZZ)
K.<a> = NumberField(x^2 + 23)
I = K.ideal(2)
P = I.factor()[0][0]
I.is_S_unit([P])

is_coprime(other)[source]¶

Return True if this ideal is coprime to other, else False.

INPUT:

other – another ideal of the same field, or generators of an ideal

OUTPUT: True if self and other are coprime, else False

Note

This function works for fractional ideals as well as integral ideals.

AUTHOR: John Cremona

EXAMPLES:

Sage

sage: x = polygen(ZZ)
sage: K.<i> = NumberField(x^2 + 1)
sage: I = K.ideal(2 + i)
sage: J = K.ideal(2 - i)
sage: I.is_coprime(J)
True
sage: (I^-1).is_coprime(J^3)
True
sage: I.is_coprime(5)
False
sage: I.is_coprime(6 + i)
True

Python

>>> from sage.all import *
>>> x = polygen(ZZ)
>>> K = NumberField(x**Integer(2) + Integer(1), names=('i',)); (i,) = K._first_ngens(1)
>>> I = K.ideal(Integer(2) + i)
>>> J = K.ideal(Integer(2) - i)
>>> I.is_coprime(J)
True
>>> (I**-Integer(1)).is_coprime(J**Integer(3))
True
>>> I.is_coprime(Integer(5))
False
>>> I.is_coprime(Integer(6) + i)
True

Sage Live

x = polygen(ZZ)
K.<i> = NumberField(x^2 + 1)
I = K.ideal(2 + i)
J = K.ideal(2 - i)
I.is_coprime(J)
(I^-1).is_coprime(J^3)
I.is_coprime(5)
I.is_coprime(6 + i)

See Issue #4536:

Sage

sage: E.<a> = NumberField(x^5 + 7*x^4 + 18*x^2 + x - 3)
sage: i,j,k = [u[0] for u in factor(3*E)]
sage: (i/j).is_coprime(j/k)
False
sage: (j/k).is_coprime(j/k)
False

sage: F.<a, b> = NumberField([x^2 - 2, x^2 - 3])
sage: F.ideal(3 - a*b).is_coprime(F.ideal(3))
False

Python

>>> from sage.all import *
>>> E = NumberField(x**Integer(5) + Integer(7)*x**Integer(4) + Integer(18)*x**Integer(2) + x - Integer(3), names=('a',)); (a,) = E._first_ngens(1)
>>> i,j,k = [u[Integer(0)] for u in factor(Integer(3)*E)]
>>> (i/j).is_coprime(j/k)
False
>>> (j/k).is_coprime(j/k)
False

>>> F = NumberField([x**Integer(2) - Integer(2), x**Integer(2) - Integer(3)], names=('a', 'b',)); (a, b,) = F._first_ngens(2)
>>> F.ideal(Integer(3) - a*b).is_coprime(F.ideal(Integer(3)))
False

Sage Live

E.<a> = NumberField(x^5 + 7*x^4 + 18*x^2 + x - 3)
i,j,k = [u[0] for u in factor(3*E)]
(i/j).is_coprime(j/k)
(j/k).is_coprime(j/k)
F.<a, b> = NumberField([x^2 - 2, x^2 - 3])
F.ideal(3 - a*b).is_coprime(F.ideal(3))

is_maximal()[source]¶

Return True if this ideal is maximal. This is equivalent to self being prime, since it is nonzero.

EXAMPLES:

Sage

sage: x = polygen(ZZ)
sage: K.<a> = NumberField(x^3 + 3); K
Number Field in a with defining polynomial x^3 + 3
sage: K.ideal(5).is_maximal()
False
sage: K.ideal(7).is_maximal()
True

Python

>>> from sage.all import *
>>> x = polygen(ZZ)
>>> K = NumberField(x**Integer(3) + Integer(3), names=('a',)); (a,) = K._first_ngens(1); K
Number Field in a with defining polynomial x^3 + 3
>>> K.ideal(Integer(5)).is_maximal()
False
>>> K.ideal(Integer(7)).is_maximal()
True

Sage Live

x = polygen(ZZ)
K.<a> = NumberField(x^3 + 3); K
K.ideal(5).is_maximal()
K.ideal(7).is_maximal()

is_trivial(proof=None)[source]¶

Return True if this is a trivial ideal.

EXAMPLES:

Sage

sage: F.<a> = QuadraticField(-5)
sage: I = F.ideal(3)
sage: I.is_trivial()
False
sage: J = F.ideal(5)
sage: J.is_trivial()
False
sage: (I + J).is_trivial()
True

Python

>>> from sage.all import *
>>> F = QuadraticField(-Integer(5), names=('a',)); (a,) = F._first_ngens(1)
>>> I = F.ideal(Integer(3))
>>> I.is_trivial()
False
>>> J = F.ideal(Integer(5))
>>> J.is_trivial()
False
>>> (I + J).is_trivial()
True

Sage Live

F.<a> = QuadraticField(-5)
I = F.ideal(3)
I.is_trivial()
J = F.ideal(5)
J.is_trivial()
(I + J).is_trivial()

numerator()[source]¶

Return the numerator ideal of this fractional ideal.

Each fractional ideal has a unique expression as \(N/D\) where \(N\), \(D\) are coprime integral ideals. The numerator is \(N\).

EXAMPLES:

Sage

sage: x = polygen(ZZ)
sage: K.<i> = NumberField(x^2 + 1)
sage: I = K.ideal((3+4*i)/5); I
Fractional ideal (4/5*i + 3/5)
sage: I.denominator()
Fractional ideal (2*i + 1)
sage: I.numerator()
Fractional ideal (-i - 2)
sage: I.numerator().is_integral() and I.denominator().is_integral()
True
sage: I.numerator() + I.denominator() == K.unit_ideal()
True
sage: I.numerator()/I.denominator() == I
True

Python

>>> from sage.all import *
>>> x = polygen(ZZ)
>>> K = NumberField(x**Integer(2) + Integer(1), names=('i',)); (i,) = K._first_ngens(1)
>>> I = K.ideal((Integer(3)+Integer(4)*i)/Integer(5)); I
Fractional ideal (4/5*i + 3/5)
>>> I.denominator()
Fractional ideal (2*i + 1)
>>> I.numerator()
Fractional ideal (-i - 2)
>>> I.numerator().is_integral() and I.denominator().is_integral()
True
>>> I.numerator() + I.denominator() == K.unit_ideal()
True
>>> I.numerator()/I.denominator() == I
True

Sage Live

x = polygen(ZZ)
K.<i> = NumberField(x^2 + 1)
I = K.ideal((3+4*i)/5); I
I.denominator()
I.numerator()
I.numerator().is_integral() and I.denominator().is_integral()
I.numerator() + I.denominator() == K.unit_ideal()
I.numerator()/I.denominator() == I

prime_factors()[source]¶

Return a list of the prime ideal factors of self.

OUTPUT:

list of prime ideals (a new list is returned each time this function is called)

EXAMPLES:

Sage

sage: x = polygen(ZZ)
sage: K.<w> = NumberField(x^2 + 23)
sage: I = ideal(w+1)
sage: I.prime_factors()
[Fractional ideal (2, 1/2*w - 1/2),
 Fractional ideal (2, 1/2*w + 1/2),
 Fractional ideal (3, 1/2*w + 1/2)]

Python

>>> from sage.all import *
>>> x = polygen(ZZ)
>>> K = NumberField(x**Integer(2) + Integer(23), names=('w',)); (w,) = K._first_ngens(1)
>>> I = ideal(w+Integer(1))
>>> I.prime_factors()
[Fractional ideal (2, 1/2*w - 1/2),
 Fractional ideal (2, 1/2*w + 1/2),
 Fractional ideal (3, 1/2*w + 1/2)]

Sage Live

x = polygen(ZZ)
K.<w> = NumberField(x^2 + 23)
I = ideal(w+1)
I.prime_factors()

prime_to_S_part(S)[source]¶

Return the part of this fractional ideal which is coprime to the prime ideals in the list \(S\).

Note

This function assumes that \(S\) is a list of prime ideals, but does not check this. This function will fail if \(S\) is not a list of prime ideals.

INPUT:

S – list of prime ideals

OUTPUT:

A fractional ideal coprime to the primes in \(S\), whose prime factorization is that of self with the primes in \(S\) removed.

EXAMPLES:

Sage

sage: x = polygen(ZZ)
sage: K.<a> = NumberField(x^2 - 23)
sage: I = K.ideal(24)
sage: S = [K.ideal(-a + 5), K.ideal(5)]
sage: I.prime_to_S_part(S)
Fractional ideal (3)
sage: J = K.ideal(15)
sage: J.prime_to_S_part(S)
Fractional ideal (3)

sage: K.<a> = NumberField(x^5 - 23)
sage: I = K.ideal(24)
sage: S = [K.ideal(15161*a^4 + 28383*a^3 + 53135*a^2 + 99478*a + 186250),
....:      K.ideal(2*a^4 + 3*a^3 + 4*a^2 + 15*a + 11),
....:      K.ideal(101)]
sage: I.prime_to_S_part(S)
Fractional ideal (24)

Python

>>> from sage.all import *
>>> x = polygen(ZZ)
>>> K = NumberField(x**Integer(2) - Integer(23), names=('a',)); (a,) = K._first_ngens(1)
>>> I = K.ideal(Integer(24))
>>> S = [K.ideal(-a + Integer(5)), K.ideal(Integer(5))]
>>> I.prime_to_S_part(S)
Fractional ideal (3)
>>> J = K.ideal(Integer(15))
>>> J.prime_to_S_part(S)
Fractional ideal (3)

>>> K = NumberField(x**Integer(5) - Integer(23), names=('a',)); (a,) = K._first_ngens(1)
>>> I = K.ideal(Integer(24))
>>> S = [K.ideal(Integer(15161)*a**Integer(4) + Integer(28383)*a**Integer(3) + Integer(53135)*a**Integer(2) + Integer(99478)*a + Integer(186250)),
...      K.ideal(Integer(2)*a**Integer(4) + Integer(3)*a**Integer(3) + Integer(4)*a**Integer(2) + Integer(15)*a + Integer(11)),
...      K.ideal(Integer(101))]
>>> I.prime_to_S_part(S)
Fractional ideal (24)

Sage Live

x = polygen(ZZ)
K.<a> = NumberField(x^2 - 23)
I = K.ideal(24)
S = [K.ideal(-a + 5), K.ideal(5)]
I.prime_to_S_part(S)
J = K.ideal(15)
J.prime_to_S_part(S)
K.<a> = NumberField(x^5 - 23)
I = K.ideal(24)
S = [K.ideal(15161*a^4 + 28383*a^3 + 53135*a^2 + 99478*a + 186250),
     K.ideal(2*a^4 + 3*a^3 + 4*a^2 + 15*a + 11),
     K.ideal(101)]
I.prime_to_S_part(S)

prime_to_idealM_part(M)[source]¶

Version for integral ideals of the prime_to_m_part function over \(\ZZ\). Return the largest divisor of self that is coprime to the ideal \(M\).

INPUT:

M – an integral ideal of the same field, or generators of an ideal

OUTPUT: an ideal which is the largest divisor of self that is coprime to \(M\)

AUTHOR: Maite Aranes

EXAMPLES:

Sage

sage: x = polygen(ZZ)
sage: k.<a> = NumberField(x^2 + 23)
sage: I = k.ideal(a + 1)
sage: M = k.ideal(2, 1/2*a - 1/2)
sage: J = I.prime_to_idealM_part(M); J
Fractional ideal (12, 1/2*a + 13/2)
sage: J.is_coprime(M)
True

sage: J = I.prime_to_idealM_part(2); J
Fractional ideal (3, 1/2*a + 1/2)
sage: J.is_coprime(M)
True

Python

>>> from sage.all import *
>>> x = polygen(ZZ)
>>> k = NumberField(x**Integer(2) + Integer(23), names=('a',)); (a,) = k._first_ngens(1)
>>> I = k.ideal(a + Integer(1))
>>> M = k.ideal(Integer(2), Integer(1)/Integer(2)*a - Integer(1)/Integer(2))
>>> J = I.prime_to_idealM_part(M); J
Fractional ideal (12, 1/2*a + 13/2)
>>> J.is_coprime(M)
True

>>> J = I.prime_to_idealM_part(Integer(2)); J
Fractional ideal (3, 1/2*a + 1/2)
>>> J.is_coprime(M)
True

Sage Live

x = polygen(ZZ)
k.<a> = NumberField(x^2 + 23)
I = k.ideal(a + 1)
M = k.ideal(2, 1/2*a - 1/2)
J = I.prime_to_idealM_part(M); J
J.is_coprime(M)
J = I.prime_to_idealM_part(2); J
J.is_coprime(M)

ramification_index()[source]¶

Return the ramification index of this fractional ideal, assuming it is prime. Otherwise, raise a ValueError.

The ramification index is the power of this prime appearing in the factorization of the prime in \(\ZZ\) that this prime lies over.

EXAMPLES:

Sage

sage: x = polygen(ZZ)
sage: K.<a> = NumberField(x^2 + 2); K
Number Field in a with defining polynomial x^2 + 2
sage: f = K.factor(2); f
(Fractional ideal (a))^2
sage: f[0][0].ramification_index()
2
sage: K.ideal(13).ramification_index()
1
sage: K.ideal(17).ramification_index()
Traceback (most recent call last):
...
ValueError: Fractional ideal (17) is not a prime ideal

Python

>>> from sage.all import *
>>> x = polygen(ZZ)
>>> K = NumberField(x**Integer(2) + Integer(2), names=('a',)); (a,) = K._first_ngens(1); K
Number Field in a with defining polynomial x^2 + 2
>>> f = K.factor(Integer(2)); f
(Fractional ideal (a))^2
>>> f[Integer(0)][Integer(0)].ramification_index()
2
>>> K.ideal(Integer(13)).ramification_index()
1
>>> K.ideal(Integer(17)).ramification_index()
Traceback (most recent call last):
...
ValueError: Fractional ideal (17) is not a prime ideal

Sage Live

x = polygen(ZZ)
K.<a> = NumberField(x^2 + 2); K
f = K.factor(2); f
f[0][0].ramification_index()
K.ideal(13).ramification_index()
K.ideal(17).ramification_index()

ray_class_number()[source]¶

Return the order of the ray class group modulo this ideal. This is a wrapper around PARI’s pari:bnrclassno function.

EXAMPLES:

Sage

sage: K.<z> = QuadraticField(-23)
sage: p = K.primes_above(3)[0]
sage: p.ray_class_number()
3

sage: x = polygen(K)
sage: L.<w> = K.extension(x^3 - z)
sage: I = L.ideal(5)
sage: I.ray_class_number()
5184

Python

>>> from sage.all import *
>>> K = QuadraticField(-Integer(23), names=('z',)); (z,) = K._first_ngens(1)
>>> p = K.primes_above(Integer(3))[Integer(0)]
>>> p.ray_class_number()
3

>>> x = polygen(K)
>>> L = K.extension(x**Integer(3) - z, names=('w',)); (w,) = L._first_ngens(1)
>>> I = L.ideal(Integer(5))
>>> I.ray_class_number()
5184

Sage Live

K.<z> = QuadraticField(-23)
p = K.primes_above(3)[0]
p.ray_class_number()
x = polygen(K)
L.<w> = K.extension(x^3 - z)
I = L.ideal(5)
I.ray_class_number()

reduce(f)[source]¶

Return the canonical reduction of the element \(f\) modulo the ideal \(I\) (= self). This is an element of \(R\) (the ring of integers of the number field) that is equivalent modulo \(I\) to \(f\).

An error is raised if this fractional ideal is not integral or the element \(f\) is not integral.

INPUT:

f – an integral element of the number field

OUTPUT:

An integral element \(g\), such that \(f - g\) belongs to the ideal self and such that \(g\) is a canonical reduced representative of the coset \(f + I\) (where \(I\) = self) as described in the method residues(), namely an integral element with coordinates \((r_0, \dots,r_{n-1})\), where:

\(r_i\) is reduced modulo \(d_i\)
\(d_i = b_i[i]\), with \(\{b_0, b_1, \dots, b_n\}\) HNF basis of the ideal self.

Note

The reduced element \(g\) is not necessarily small. To get a small \(g\) use the method small_residue().

EXAMPLES:

Sage

sage: x = polygen(ZZ)
sage: k.<a> = NumberField(x^3 + 11)
sage: I = k.ideal(5, a^2 - a + 1)
sage: c = 4*a + 9
sage: I.reduce(c)
a^2 - 2*a
sage: c - I.reduce(c) in I
True

Python

>>> from sage.all import *
>>> x = polygen(ZZ)
>>> k = NumberField(x**Integer(3) + Integer(11), names=('a',)); (a,) = k._first_ngens(1)
>>> I = k.ideal(Integer(5), a**Integer(2) - a + Integer(1))
>>> c = Integer(4)*a + Integer(9)
>>> I.reduce(c)
a^2 - 2*a
>>> c - I.reduce(c) in I
True

Sage Live

x = polygen(ZZ)
k.<a> = NumberField(x^3 + 11)
I = k.ideal(5, a^2 - a + 1)
c = 4*a + 9
I.reduce(c)
c - I.reduce(c) in I

The reduced element is in the list of canonical representatives returned by the residues method:

Sage

sage: I.reduce(c) in list(I.residues())
True

Python

>>> from sage.all import *
>>> I.reduce(c) in list(I.residues())
True

Sage Live

I.reduce(c) in list(I.residues())

The reduced element does not necessarily have smaller norm (use small_residue() for that)

Sage

sage: c.norm()
25
sage: (I.reduce(c)).norm()
209
sage: (I.small_residue(c)).norm()
10

Python

>>> from sage.all import *
>>> c.norm()
25
>>> (I.reduce(c)).norm()
209
>>> (I.small_residue(c)).norm()
10

Sage Live

c.norm()
(I.reduce(c)).norm()
(I.small_residue(c)).norm()

Sometimes the canonical reduced representative of \(1\) won’t be \(1\) (it depends on the choice of basis for the ring of integers):

Sage

sage: k.<a> = NumberField(x^2 + 23)
sage: I = k.ideal(3)
sage: I.reduce(3*a + 1)
-3/2*a - 1/2
sage: k.ring_of_integers().basis()
[1/2*a + 1/2, a]

Python

>>> from sage.all import *
>>> k = NumberField(x**Integer(2) + Integer(23), names=('a',)); (a,) = k._first_ngens(1)
>>> I = k.ideal(Integer(3))
>>> I.reduce(Integer(3)*a + Integer(1))
-3/2*a - 1/2
>>> k.ring_of_integers().basis()
[1/2*a + 1/2, a]

Sage Live

k.<a> = NumberField(x^2 + 23)
I = k.ideal(3)
I.reduce(3*a + 1)
k.ring_of_integers().basis()

AUTHOR: Maite Aranes.

residue_class_degree()[source]¶

Return the residue class degree of this fractional ideal, assuming it is prime. Otherwise, raise a ValueError.

The residue class degree of a prime ideal \(I\) is the degree of the extension \(O_K/I\) of its prime subfield.

EXAMPLES:

Sage

sage: x = polygen(ZZ)
sage: K.<a> = NumberField(x^5 + 2); K
Number Field in a with defining polynomial x^5 + 2
sage: f = K.factor(19); f
(Fractional ideal (a^2 + a - 3))
 * (Fractional ideal (2*a^4 + a^2 - 2*a + 1))
 * (Fractional ideal (a^2 + a - 1))
sage: [i.residue_class_degree() for i, _ in f]
[2, 2, 1]

Python

>>> from sage.all import *
>>> x = polygen(ZZ)
>>> K = NumberField(x**Integer(5) + Integer(2), names=('a',)); (a,) = K._first_ngens(1); K
Number Field in a with defining polynomial x^5 + 2
>>> f = K.factor(Integer(19)); f
(Fractional ideal (a^2 + a - 3))
 * (Fractional ideal (2*a^4 + a^2 - 2*a + 1))
 * (Fractional ideal (a^2 + a - 1))
>>> [i.residue_class_degree() for i, _ in f]
[2, 2, 1]

Sage Live

x = polygen(ZZ)
K.<a> = NumberField(x^5 + 2); K
f = K.factor(19); f
[i.residue_class_degree() for i, _ in f]

residue_field(names=None)[source]¶

Return the residue class field of this fractional ideal, which must be prime.

EXAMPLES:

Sage

sage: x = polygen(ZZ)
sage: K.<a> = NumberField(x^3 - 7)
sage: P = K.ideal(29).factor()[0][0]
sage: P.residue_field()
Residue field in abar of Fractional ideal (2*a^2 + 3*a - 10)
sage: P.residue_field('z')
Residue field in z of Fractional ideal (2*a^2 + 3*a - 10)

Python

>>> from sage.all import *
>>> x = polygen(ZZ)
>>> K = NumberField(x**Integer(3) - Integer(7), names=('a',)); (a,) = K._first_ngens(1)
>>> P = K.ideal(Integer(29)).factor()[Integer(0)][Integer(0)]
>>> P.residue_field()
Residue field in abar of Fractional ideal (2*a^2 + 3*a - 10)
>>> P.residue_field('z')
Residue field in z of Fractional ideal (2*a^2 + 3*a - 10)

Sage Live

x = polygen(ZZ)
K.<a> = NumberField(x^3 - 7)
P = K.ideal(29).factor()[0][0]
P.residue_field()
P.residue_field('z')

Another example:

Sage

sage: K.<a> = NumberField(x^3 - 7)
sage: P = K.ideal(389).factor()[0][0]; P
Fractional ideal (389, a^2 - 44*a - 9)
sage: P.residue_class_degree()
2
sage: P.residue_field()
Residue field in abar of Fractional ideal (389, a^2 - 44*a - 9)
sage: P.residue_field('z')
Residue field in z of Fractional ideal (389, a^2 - 44*a - 9)
sage: FF.<w> = P.residue_field()
sage: FF
Residue field in w of Fractional ideal (389, a^2 - 44*a - 9)
sage: FF((a+1)^390)
36
sage: FF(a)
w

Python

>>> from sage.all import *
>>> K = NumberField(x**Integer(3) - Integer(7), names=('a',)); (a,) = K._first_ngens(1)
>>> P = K.ideal(Integer(389)).factor()[Integer(0)][Integer(0)]; P
Fractional ideal (389, a^2 - 44*a - 9)
>>> P.residue_class_degree()
2
>>> P.residue_field()
Residue field in abar of Fractional ideal (389, a^2 - 44*a - 9)
>>> P.residue_field('z')
Residue field in z of Fractional ideal (389, a^2 - 44*a - 9)
>>> FF = P.residue_field(names=('w',)); (w,) = FF._first_ngens(1)
>>> FF
Residue field in w of Fractional ideal (389, a^2 - 44*a - 9)
>>> FF((a+Integer(1))**Integer(390))
36
>>> FF(a)
w

Sage Live

K.<a> = NumberField(x^3 - 7)
P = K.ideal(389).factor()[0][0]; P
P.residue_class_degree()
P.residue_field()
P.residue_field('z')
FF.<w> = P.residue_field()
FF
FF((a+1)^390)
FF(a)

An example of reduction maps to the residue field: these are defined on the whole valuation ring, i.e. the subring of the number field consisting of elements with nonnegative valuation. This shows that the issue raised in Issue #1951 has been fixed:

Sage

sage: K.<i> = NumberField(x^2 + 1)
sage: P1, P2 = [g[0] for g in K.factor(5)]; P1, P2
(Fractional ideal (-i - 2), Fractional ideal (2*i + 1))
sage: a = 1/(1+2*i)
sage: F1, F2 = [g.residue_field() for g in [P1, P2]]; F1, F2
(Residue field of Fractional ideal (-i - 2),
 Residue field of Fractional ideal (2*i + 1))
sage: a.valuation(P1)
0
sage: F1(i/7)
4
sage: F1(a)
3
sage: a.valuation(P2)
-1
sage: F2(a)
Traceback (most recent call last):
...
ZeroDivisionError: Cannot reduce field element -2/5*i + 1/5
modulo Fractional ideal (2*i + 1): it has negative valuation

Python

>>> from sage.all import *
>>> K = NumberField(x**Integer(2) + Integer(1), names=('i',)); (i,) = K._first_ngens(1)
>>> P1, P2 = [g[Integer(0)] for g in K.factor(Integer(5))]; P1, P2
(Fractional ideal (-i - 2), Fractional ideal (2*i + 1))
>>> a = Integer(1)/(Integer(1)+Integer(2)*i)
>>> F1, F2 = [g.residue_field() for g in [P1, P2]]; F1, F2
(Residue field of Fractional ideal (-i - 2),
 Residue field of Fractional ideal (2*i + 1))
>>> a.valuation(P1)
0
>>> F1(i/Integer(7))
4
>>> F1(a)
3
>>> a.valuation(P2)
-1
>>> F2(a)
Traceback (most recent call last):
...
ZeroDivisionError: Cannot reduce field element -2/5*i + 1/5
modulo Fractional ideal (2*i + 1): it has negative valuation

Sage Live

K.<i> = NumberField(x^2 + 1)
P1, P2 = [g[0] for g in K.factor(5)]; P1, P2
a = 1/(1+2*i)
F1, F2 = [g.residue_field() for g in [P1, P2]]; F1, F2
a.valuation(P1)
F1(i/7)
F1(a)
a.valuation(P2)
F2(a)

An example with a relative number field:

Sage

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: R = p.residue_field(); R
Residue field in abar of Fractional ideal ((-1/2*b - 1/2)*a + 1/2*b - 1/2)
sage: R.cardinality()
9
sage: R(17)
2
sage: R((a + b)/17)
abar
sage: R(1/b)
2*abar

Python

>>> from sage.all import *
>>> L = NumberField([x**Integer(2) + Integer(1), x**Integer(2) - Integer(5)], names=('a', 'b',)); (a, b,) = L._first_ngens(2)
>>> p = L.ideal((-Integer(1)/Integer(2)*b - Integer(1)/Integer(2))*a + Integer(1)/Integer(2)*b - Integer(1)/Integer(2))
>>> R = p.residue_field(); R
Residue field in abar of Fractional ideal ((-1/2*b - 1/2)*a + 1/2*b - 1/2)
>>> R.cardinality()
9
>>> R(Integer(17))
2
>>> R((a + b)/Integer(17))
abar
>>> R(Integer(1)/b)
2*abar

Sage Live

L.<a,b> = NumberField([x^2 + 1, x^2 - 5])
p = L.ideal((-1/2*b - 1/2)*a + 1/2*b - 1/2)
R = p.residue_field(); R
R.cardinality()
R(17)
R((a + b)/17)
R(1/b)

We verify that Issue #8721 is fixed:

Sage

sage: L.<a, b> = NumberField([x^2 - 3, x^2 - 5])
sage: L.ideal(a).residue_field()
Residue field in abar of Fractional ideal (a)

Python

>>> from sage.all import *
>>> L = NumberField([x**Integer(2) - Integer(3), x**Integer(2) - Integer(5)], names=('a', 'b',)); (a, b,) = L._first_ngens(2)
>>> L.ideal(a).residue_field()
Residue field in abar of Fractional ideal (a)

Sage Live

L.<a, b> = NumberField([x^2 - 3, x^2 - 5])
L.ideal(a).residue_field()

residues()[source]¶

Return a iterator through a complete list of residues modulo this integral ideal.

An error is raised if this fractional ideal is not integral.

OUTPUT:

An iterator through a complete list of residues modulo the integral ideal self. This list is the set of canonical reduced representatives given by all integral elements with coordinates \((r_0, \dots,r_{n-1})\), where:

\(r_i\) is reduced modulo \(d_i\)
\(d_i = b_i[i]\), with \(\{b_0, b_1, \dots, b_n\}\) HNF basis of the ideal.

AUTHOR: John Cremona (modified by Maite Aranes)

EXAMPLES:

Sage

sage: x = polygen(ZZ)
sage: K.<i> = NumberField(x^2 + 1)
sage: res = K.ideal(2).residues(); res
xmrange_iter([[0, 1], [0, 1]], <function ...<lambda> at 0x...>)
sage: list(res)
[0, i, 1, i + 1]
sage: list(K.ideal(2 + i).residues())
[-2*i, -i, 0, i, 2*i]
sage: list(K.ideal(i).residues())
[0]
sage: I = K.ideal(3 + 6*i)
sage: reps = I.residues()
sage: len(list(reps)) == I.norm()
True
sage: all(r == s or not (r-s) in I      # long time (6s on sage.math, 2011)
....:     for r in reps for s in reps)
True

sage: K.<a> = NumberField(x^3 - 10)
sage: I = K.ideal(a - 1)
sage: len(list(I.residues())) == I.norm()
True

sage: K.<z> = CyclotomicField(11)
sage: len(list(K.primes_above(3)[0].residues())) == 3**5  # long time (5s on sage.math, 2011)
True

Python

>>> from sage.all import *
>>> x = polygen(ZZ)
>>> K = NumberField(x**Integer(2) + Integer(1), names=('i',)); (i,) = K._first_ngens(1)
>>> res = K.ideal(Integer(2)).residues(); res
xmrange_iter([[0, 1], [0, 1]], <function ...<lambda> at 0x...>)
>>> list(res)
[0, i, 1, i + 1]
>>> list(K.ideal(Integer(2) + i).residues())
[-2*i, -i, 0, i, 2*i]
>>> list(K.ideal(i).residues())
[0]
>>> I = K.ideal(Integer(3) + Integer(6)*i)
>>> reps = I.residues()
>>> len(list(reps)) == I.norm()
True
>>> all(r == s or not (r-s) in I      # long time (6s on sage.math, 2011)
...     for r in reps for s in reps)
True

>>> K = NumberField(x**Integer(3) - Integer(10), names=('a',)); (a,) = K._first_ngens(1)
>>> I = K.ideal(a - Integer(1))
>>> len(list(I.residues())) == I.norm()
True

>>> K = CyclotomicField(Integer(11), names=('z',)); (z,) = K._first_ngens(1)
>>> len(list(K.primes_above(Integer(3))[Integer(0)].residues())) == Integer(3)**Integer(5)  # long time (5s on sage.math, 2011)
True

Sage Live

x = polygen(ZZ)
K.<i> = NumberField(x^2 + 1)
res = K.ideal(2).residues(); res
list(res)
list(K.ideal(2 + i).residues())
list(K.ideal(i).residues())
I = K.ideal(3 + 6*i)
reps = I.residues()
len(list(reps)) == I.norm()
all(r == s or not (r-s) in I      # long time (6s on sage.math, 2011)
    for r in reps for s in reps)
K.<a> = NumberField(x^3 - 10)
I = K.ideal(a - 1)
len(list(I.residues())) == I.norm()
K.<z> = CyclotomicField(11)
len(list(K.primes_above(3)[0].residues())) == 3**5  # long time (5s on sage.math, 2011)

small_residue(f)[source]¶

Given an element \(f\) of the ambient number field, return an element \(g\) such that \(f - g\) belongs to the ideal self (which must be integral), and \(g\) is small.

Note

The reduced representative returned is not uniquely determined.

ALGORITHM: Uses PARI function pari:nfeltreduce.

EXAMPLES:

Sage

sage: x = polygen(ZZ)
sage: k.<a> = NumberField(x^2 + 5)
sage: I = k.ideal(a)
sage: I.small_residue(14)
4

Python

>>> from sage.all import *
>>> x = polygen(ZZ)
>>> k = NumberField(x**Integer(2) + Integer(5), names=('a',)); (a,) = k._first_ngens(1)
>>> I = k.ideal(a)
>>> I.small_residue(Integer(14))
4

Sage Live

x = polygen(ZZ)
k.<a> = NumberField(x^2 + 5)
I = k.ideal(a)
I.small_residue(14)

Sage

sage: K.<a> = NumberField(x^5 + 7*x^4 + 18*x^2 + x - 3)
sage: I = K.ideal(5)
sage: I.small_residue(a^2 -13)
a^2 + 5*a - 3

Python

>>> from sage.all import *
>>> K = NumberField(x**Integer(5) + Integer(7)*x**Integer(4) + Integer(18)*x**Integer(2) + x - Integer(3), names=('a',)); (a,) = K._first_ngens(1)
>>> I = K.ideal(Integer(5))
>>> I.small_residue(a**Integer(2) -Integer(13))
a^2 + 5*a - 3

Sage Live

K.<a> = NumberField(x^5 + 7*x^4 + 18*x^2 + x - 3)
I = K.ideal(5)
I.small_residue(a^2 -13)

Python

>>> from sage.all import *
>>> K = NumberField(x**Integer(5) + Integer(7)*x**Integer(4) + Integer(18)*x**Integer(2) + x - Integer(3), names=('a',)); (a,) = K._first_ngens(1)
>>> I = K.ideal(Integer(5))
>>> I.small_residue(a**Integer(2) -Integer(13))
a^2 + 5*a - 3

Sage Live

K.<a> = NumberField(x^5 + 7*x^4 + 18*x^2 + x - 3)
I = K.ideal(5)
I.small_residue(a^2 -13)

support()[source]¶

Return a list of the prime ideal factors of self.

OUTPUT:

list of prime ideals (a new list is returned each time this function is called)

EXAMPLES:

Sage

sage: x = polygen(ZZ)
sage: K.<w> = NumberField(x^2 + 23)
sage: I = ideal(w+1)
sage: I.prime_factors()
[Fractional ideal (2, 1/2*w - 1/2),
 Fractional ideal (2, 1/2*w + 1/2),
 Fractional ideal (3, 1/2*w + 1/2)]

Python

>>> from sage.all import *
>>> x = polygen(ZZ)
>>> K = NumberField(x**Integer(2) + Integer(23), names=('w',)); (w,) = K._first_ngens(1)
>>> I = ideal(w+Integer(1))
>>> I.prime_factors()
[Fractional ideal (2, 1/2*w - 1/2),
 Fractional ideal (2, 1/2*w + 1/2),
 Fractional ideal (3, 1/2*w + 1/2)]

Sage Live

x = polygen(ZZ)
K.<w> = NumberField(x^2 + 23)
I = ideal(w+1)
I.prime_factors()

class sage.rings.number_field.number_field_ideal.NumberFieldIdeal(field, gens, coerce=True)[source]¶

Bases: Ideal_generic

An ideal of a number field.

EXAMPLES:

Sage

sage: x = polygen(ZZ)
sage: K.<i> = NumberField(x^2 + 1)
sage: K.ideal(7)
Fractional ideal (7)

Python

>>> from sage.all import *
>>> x = polygen(ZZ)
>>> K = NumberField(x**Integer(2) + Integer(1), names=('i',)); (i,) = K._first_ngens(1)
>>> K.ideal(Integer(7))
Fractional ideal (7)

Sage Live

x = polygen(ZZ)
K.<i> = NumberField(x^2 + 1)
K.ideal(7)

Initialization from PARI:

Sage

sage: K.ideal(pari(7))
Fractional ideal (7)
sage: K.ideal(pari(4), pari(4 + 2*i))
Fractional ideal (2)
sage: K.ideal(pari("i + 2"))
Fractional ideal (i + 2)
sage: K.ideal(pari("[3,0;0,3]"))
Fractional ideal (3)
sage: F = pari(K).idealprimedec(5)
sage: K.ideal(F[0])
Fractional ideal (2*i + 1)

Python

>>> from sage.all import *
>>> K.ideal(pari(Integer(7)))
Fractional ideal (7)
>>> K.ideal(pari(Integer(4)), pari(Integer(4) + Integer(2)*i))
Fractional ideal (2)
>>> K.ideal(pari("i + 2"))
Fractional ideal (i + 2)
>>> K.ideal(pari("[3,0;0,3]"))
Fractional ideal (3)
>>> F = pari(K).idealprimedec(Integer(5))
>>> K.ideal(F[Integer(0)])
Fractional ideal (2*i + 1)

Sage Live

K.ideal(pari(7))
K.ideal(pari(4), pari(4 + 2*i))
K.ideal(pari("i + 2"))
K.ideal(pari("[3,0;0,3]"))
F = pari(K).idealprimedec(5)
K.ideal(F[0])

S_ideal_class_log(S)[source]¶

S-class group version of ideal_class_log().

EXAMPLES:

Sage

sage: K.<a> = QuadraticField(-14)
sage: S = K.primes_above(2)
sage: I = K.ideal(3, a + 1)
sage: I.S_ideal_class_log(S)
[1]
sage: I.S_ideal_class_log([])
[3]

Python

>>> from sage.all import *
>>> K = QuadraticField(-Integer(14), names=('a',)); (a,) = K._first_ngens(1)
>>> S = K.primes_above(Integer(2))
>>> I = K.ideal(Integer(3), a + Integer(1))
>>> I.S_ideal_class_log(S)
[1]
>>> I.S_ideal_class_log([])
[3]

Sage Live

K.<a> = QuadraticField(-14)
S = K.primes_above(2)
I = K.ideal(3, a + 1)
I.S_ideal_class_log(S)
I.S_ideal_class_log([])

absolute_norm()[source]¶

A synonym for norm().

EXAMPLES:

Sage

sage: x = polygen(ZZ)
sage: K.<i> = NumberField(x^2 + 1)
sage: K.ideal(1 + 2*i).absolute_norm()
5

Python

>>> from sage.all import *
>>> x = polygen(ZZ)
>>> K = NumberField(x**Integer(2) + Integer(1), names=('i',)); (i,) = K._first_ngens(1)
>>> K.ideal(Integer(1) + Integer(2)*i).absolute_norm()
5

Sage Live

x = polygen(ZZ)
K.<i> = NumberField(x^2 + 1)
K.ideal(1 + 2*i).absolute_norm()

absolute_ramification_index()[source]¶

A synonym for ramification_index().

EXAMPLES:

Sage

sage: x = polygen(ZZ)
sage: K.<i> = NumberField(x^2 + 1)
sage: K.ideal(1 + i).absolute_ramification_index()
2

Python

>>> from sage.all import *
>>> x = polygen(ZZ)
>>> K = NumberField(x**Integer(2) + Integer(1), names=('i',)); (i,) = K._first_ngens(1)
>>> K.ideal(Integer(1) + i).absolute_ramification_index()
2

Sage Live

x = polygen(ZZ)
K.<i> = NumberField(x^2 + 1)
K.ideal(1 + i).absolute_ramification_index()

artin_symbol()[source]¶

Return the Artin symbol \((K / \QQ, P)\), where \(K\) is the number field of \(P\) = self. This is the unique element \(s\) of the decomposition group of \(P\) such that \(s(x) = x^p \pmod{P}\) where \(p\) is the residue characteristic of \(P\). (Here \(P\) (self) should be prime and unramified.)

See the GaloisGroup_v2.artin_symbol() method for further documentation and examples.

EXAMPLES:

Sage

sage: QuadraticField(-23, 'w').primes_above(7)[0].artin_symbol()            # needs sage.groups
(1,2)

Python

>>> from sage.all import *
>>> QuadraticField(-Integer(23), 'w').primes_above(Integer(7))[Integer(0)].artin_symbol()            # needs sage.groups
(1,2)

Sage Live

QuadraticField(-23, 'w').primes_above(7)[0].artin_symbol()            # needs sage.groups

basis()[source]¶

Return a basis for this ideal viewed as a \(\ZZ\)-module.

OUTPUT:

An immutable sequence of elements of this ideal (note: their parent is the number field) forming a basis for this ideal.

EXAMPLES:

Sage

sage: K.<z> = CyclotomicField(7)
sage: I = K.factor(11)[0][0]
sage: I.basis()           # warning -- choice of basis can be somewhat random
[11, 11*z, 11*z^2, z^3 + 5*z^2 + 4*z + 10,
 z^4 + z^2 + z + 5, z^5 + z^4 + z^3 + 2*z^2 + 6*z + 5]

Python

>>> from sage.all import *
>>> K = CyclotomicField(Integer(7), names=('z',)); (z,) = K._first_ngens(1)
>>> I = K.factor(Integer(11))[Integer(0)][Integer(0)]
>>> I.basis()           # warning -- choice of basis can be somewhat random
[11, 11*z, 11*z^2, z^3 + 5*z^2 + 4*z + 10,
 z^4 + z^2 + z + 5, z^5 + z^4 + z^3 + 2*z^2 + 6*z + 5]

Sage Live

K.<z> = CyclotomicField(7)
I = K.factor(11)[0][0]
I.basis()           # warning -- choice of basis can be somewhat random

An example of a non-integral ideal.:

Sage

sage: J = 1/I
sage: J          # warning -- choice of generators can be somewhat random
Fractional ideal (2/11*z^5 + 2/11*z^4 + 3/11*z^3 + 2/11)
sage: J.basis()           # warning -- choice of basis can be somewhat random
[1, z, z^2, 1/11*z^3 + 7/11*z^2 + 6/11*z + 10/11,
 1/11*z^4 + 1/11*z^2 + 1/11*z + 7/11,
 1/11*z^5 + 1/11*z^4 + 1/11*z^3 + 2/11*z^2 + 8/11*z + 7/11]

Python

>>> from sage.all import *
>>> J = Integer(1)/I
>>> J          # warning -- choice of generators can be somewhat random
Fractional ideal (2/11*z^5 + 2/11*z^4 + 3/11*z^3 + 2/11)
>>> J.basis()           # warning -- choice of basis can be somewhat random
[1, z, z^2, 1/11*z^3 + 7/11*z^2 + 6/11*z + 10/11,
 1/11*z^4 + 1/11*z^2 + 1/11*z + 7/11,
 1/11*z^5 + 1/11*z^4 + 1/11*z^3 + 2/11*z^2 + 8/11*z + 7/11]

Sage Live

J = 1/I
J          # warning -- choice of generators can be somewhat random
J.basis()           # warning -- choice of basis can be somewhat random

Number fields defined by non-monic and non-integral polynomials are supported (Issue #252):

Sage

sage: x = polygen(ZZ)
sage: K.<a> = NumberField(2*x^2 - 1/3)
sage: K.ideal(a).basis()
[1, a]

Python

>>> from sage.all import *
>>> x = polygen(ZZ)
>>> K = NumberField(Integer(2)*x**Integer(2) - Integer(1)/Integer(3), names=('a',)); (a,) = K._first_ngens(1)
>>> K.ideal(a).basis()
[1, a]

Sage Live

x = polygen(ZZ)
K.<a> = NumberField(2*x^2 - 1/3)
K.ideal(a).basis()

coordinates(x)[source]¶

Return the coordinate vector of \(x\) with respect to this ideal.

INPUT:

x – an element of the number field (or ring of integers) of this ideal

OUTPUT:

List giving the coordinates of \(x\) with respect to the integral basis of the ideal. In general this will be a vector of rationals; it will consist of integers if and only if \(x\) is in the ideal.

AUTHOR: John Cremona 2008-10-31

ALGORITHM:

Uses linear algebra. Provides simpler implementations for _contains_(), is_integral() and smallest_integer().

EXAMPLES:

Sage

sage: K.<i> = QuadraticField(-1)
sage: I = K.ideal(7 + 3*i)
sage: Ibasis = I.integral_basis(); Ibasis
[58, i + 41]
sage: a = 23 - 14*i
sage: acoords = I.coordinates(a); acoords
(597/58, -14)
sage: sum([Ibasis[j]*acoords[j] for j in range(2)]) == a
True
sage: b = 123 + 456*i
sage: bcoords = I.coordinates(b); bcoords
(-18573/58, 456)
sage: sum([Ibasis[j]*bcoords[j] for j in range(2)]) == b
True
sage: J = K.ideal(0)
sage: J.coordinates(0)
()
sage: J.coordinates(1)
Traceback (most recent call last):
...
TypeError: vector is not in free module

Python

>>> from sage.all import *
>>> K = QuadraticField(-Integer(1), names=('i',)); (i,) = K._first_ngens(1)
>>> I = K.ideal(Integer(7) + Integer(3)*i)
>>> Ibasis = I.integral_basis(); Ibasis
[58, i + 41]
>>> a = Integer(23) - Integer(14)*i
>>> acoords = I.coordinates(a); acoords
(597/58, -14)
>>> sum([Ibasis[j]*acoords[j] for j in range(Integer(2))]) == a
True
>>> b = Integer(123) + Integer(456)*i
>>> bcoords = I.coordinates(b); bcoords
(-18573/58, 456)
>>> sum([Ibasis[j]*bcoords[j] for j in range(Integer(2))]) == b
True
>>> J = K.ideal(Integer(0))
>>> J.coordinates(Integer(0))
()
>>> J.coordinates(Integer(1))
Traceback (most recent call last):
...
TypeError: vector is not in free module

Sage Live

K.<i> = QuadraticField(-1)
I = K.ideal(7 + 3*i)
Ibasis = I.integral_basis(); Ibasis
a = 23 - 14*i
acoords = I.coordinates(a); acoords
sum([Ibasis[j]*acoords[j] for j in range(2)]) == a
b = 123 + 456*i
bcoords = I.coordinates(b); bcoords
sum([Ibasis[j]*bcoords[j] for j in range(2)]) == b
J = K.ideal(0)
J.coordinates(0)
J.coordinates(1)

decomposition_group()[source]¶

Return the decomposition group of self, as a subset of the automorphism group of the number field of self. Raises an error if the field isn’t Galois. See the GaloisGroup_v2.decomposition_group() method for further examples and doctests.

EXAMPLES:

Sage

sage: QuadraticField(-23, 'w').primes_above(7)[0].decomposition_group()     # needs sage.groups
Subgroup generated by [(1,2)] of (Galois group 2T1 (S2) with order 2 of x^2 + 23)

Python

>>> from sage.all import *
>>> QuadraticField(-Integer(23), 'w').primes_above(Integer(7))[Integer(0)].decomposition_group()     # needs sage.groups
Subgroup generated by [(1,2)] of (Galois group 2T1 (S2) with order 2 of x^2 + 23)

Sage Live

QuadraticField(-23, 'w').primes_above(7)[0].decomposition_group()     # needs sage.groups

free_module()[source]¶

Return the free \(\ZZ\)-module contained in the vector space associated to the ambient number field, that corresponds to this ideal.

EXAMPLES:

Sage

sage: K.<z> = CyclotomicField(7)
sage: I = K.factor(11)[0][0]; I
Fractional ideal (-3*z^4 - 2*z^3 - 2*z^2 - 2)
sage: A = I.free_module()
sage: A              # warning -- choice of basis can be somewhat random
Free module of degree 6 and rank 6 over Integer Ring
User basis matrix:
[11  0  0  0  0  0]
[ 0 11  0  0  0  0]
[ 0  0 11  0  0  0]
[10  4  5  1  0  0]
[ 5  1  1  0  1  0]
[ 5  6  2  1  1  1]

Python

>>> from sage.all import *
>>> K = CyclotomicField(Integer(7), names=('z',)); (z,) = K._first_ngens(1)
>>> I = K.factor(Integer(11))[Integer(0)][Integer(0)]; I
Fractional ideal (-3*z^4 - 2*z^3 - 2*z^2 - 2)
>>> A = I.free_module()
>>> A              # warning -- choice of basis can be somewhat random
Free module of degree 6 and rank 6 over Integer Ring
User basis matrix:
[11  0  0  0  0  0]
[ 0 11  0  0  0  0]
[ 0  0 11  0  0  0]
[10  4  5  1  0  0]
[ 5  1  1  0  1  0]
[ 5  6  2  1  1  1]

Sage Live

K.<z> = CyclotomicField(7)
I = K.factor(11)[0][0]; I
A = I.free_module()
A              # warning -- choice of basis can be somewhat random

However, the actual \(\ZZ\)-module is not at all random:

Sage

sage: A.basis_matrix().change_ring(ZZ).echelon_form()
[ 1  0  0  5  1  1]
[ 0  1  0  1  1  7]
[ 0  0  1  7  6 10]
[ 0  0  0 11  0  0]
[ 0  0  0  0 11  0]
[ 0  0  0  0  0 11]

Python

>>> from sage.all import *
>>> A.basis_matrix().change_ring(ZZ).echelon_form()
[ 1  0  0  5  1  1]
[ 0  1  0  1  1  7]
[ 0  0  1  7  6 10]
[ 0  0  0 11  0  0]
[ 0  0  0  0 11  0]
[ 0  0  0  0  0 11]

Sage Live

A.basis_matrix().change_ring(ZZ).echelon_form()

The ideal doesn’t have to be integral:

Sage

sage: J = I^(-1)
sage: B = J.free_module()
sage: B.echelonized_basis_matrix()
[ 1/11     0     0  7/11  1/11  1/11]
[    0  1/11     0  1/11  1/11  5/11]
[    0     0  1/11  5/11  4/11 10/11]
[    0     0     0     1     0     0]
[    0     0     0     0     1     0]
[    0     0     0     0     0     1]

Python

>>> from sage.all import *
>>> J = I**(-Integer(1))
>>> B = J.free_module()
>>> B.echelonized_basis_matrix()
[ 1/11     0     0  7/11  1/11  1/11]
[    0  1/11     0  1/11  1/11  5/11]
[    0     0  1/11  5/11  4/11 10/11]
[    0     0     0     1     0     0]
[    0     0     0     0     1     0]
[    0     0     0     0     0     1]

Sage Live

J = I^(-1)
B = J.free_module()
B.echelonized_basis_matrix()

This also works for relative extensions:

Sage

sage: x = polygen(ZZ)
sage: K.<a,b> = NumberField([x^2 + 1, x^2 + 2])
sage: I = K.fractional_ideal(4)
sage: I.free_module()
Free module of degree 4 and rank 4 over Integer Ring
User basis matrix:
[  4   0   0   0]
[ -3   7  -1   1]
[  3   7   1   1]
[  0 -10   0  -2]
sage: J = I^(-1); J.free_module()
Free module of degree 4 and rank 4 over Integer Ring
User basis matrix:
[  1/4     0     0     0]
[-3/16  7/16 -1/16  1/16]
[ 3/16  7/16  1/16  1/16]
[    0  -5/8     0  -1/8]

Python

>>> from sage.all import *
>>> x = polygen(ZZ)
>>> K = NumberField([x**Integer(2) + Integer(1), x**Integer(2) + Integer(2)], names=('a', 'b',)); (a, b,) = K._first_ngens(2)
>>> I = K.fractional_ideal(Integer(4))
>>> I.free_module()
Free module of degree 4 and rank 4 over Integer Ring
User basis matrix:
[  4   0   0   0]
[ -3   7  -1   1]
[  3   7   1   1]
[  0 -10   0  -2]
>>> J = I**(-Integer(1)); J.free_module()
Free module of degree 4 and rank 4 over Integer Ring
User basis matrix:
[  1/4     0     0     0]
[-3/16  7/16 -1/16  1/16]
[ 3/16  7/16  1/16  1/16]
[    0  -5/8     0  -1/8]

Sage Live

x = polygen(ZZ)
K.<a,b> = NumberField([x^2 + 1, x^2 + 2])
I = K.fractional_ideal(4)
I.free_module()
J = I^(-1); J.free_module()

An example of intersecting ideals by intersecting free modules.:

Sage

sage: K.<a> = NumberField(x^3 + x^2 - 2*x + 8)
sage: I = K.factor(2)
sage: p1 = I[0][0]; p2 = I[1][0]
sage: N = p1.free_module().intersection(p2.free_module()); N
Free module of degree 3 and rank 3 over Integer Ring
Echelon basis matrix:
[  1 1/2 1/2]
[  0   1   1]
[  0   0   2]
sage: N.index_in(p1.free_module()).abs()
2

Python

>>> from sage.all import *
>>> K = NumberField(x**Integer(3) + x**Integer(2) - Integer(2)*x + Integer(8), names=('a',)); (a,) = K._first_ngens(1)
>>> I = K.factor(Integer(2))
>>> p1 = I[Integer(0)][Integer(0)]; p2 = I[Integer(1)][Integer(0)]
>>> N = p1.free_module().intersection(p2.free_module()); N
Free module of degree 3 and rank 3 over Integer Ring
Echelon basis matrix:
[  1 1/2 1/2]
[  0   1   1]
[  0   0   2]
>>> N.index_in(p1.free_module()).abs()
2

Sage Live

K.<a> = NumberField(x^3 + x^2 - 2*x + 8)
I = K.factor(2)
p1 = I[0][0]; p2 = I[1][0]
N = p1.free_module().intersection(p2.free_module()); N
N.index_in(p1.free_module()).abs()

gens_reduced(proof=None)[source]¶

Express this ideal in terms of at most two generators, and one if possible.

This function indirectly uses pari:bnfisprincipal, so set proof=True if you want to prove correctness (which is the default).

EXAMPLES:

Sage

sage: x = polygen(ZZ)
sage: R.<x> = PolynomialRing(QQ)
sage: K.<a> = NumberField(x^2 + 5)
sage: K.ideal(0).gens_reduced()
(0,)
sage: J = K.ideal([a + 2, 9])
sage: J.gens()
(a + 2, 9)
sage: J.gens_reduced()  # random sign
(a + 2,)
sage: K.ideal([a + 2, 3]).gens_reduced()
(3, a + 2)

Python

>>> from sage.all import *
>>> x = polygen(ZZ)
>>> R = PolynomialRing(QQ, names=('x',)); (x,) = R._first_ngens(1)
>>> K = NumberField(x**Integer(2) + Integer(5), names=('a',)); (a,) = K._first_ngens(1)
>>> K.ideal(Integer(0)).gens_reduced()
(0,)
>>> J = K.ideal([a + Integer(2), Integer(9)])
>>> J.gens()
(a + 2, 9)
>>> J.gens_reduced()  # random sign
(a + 2,)
>>> K.ideal([a + Integer(2), Integer(3)]).gens_reduced()
(3, a + 2)

Sage Live

x = polygen(ZZ)
R.<x> = PolynomialRing(QQ)
K.<a> = NumberField(x^2 + 5)
K.ideal(0).gens_reduced()
J = K.ideal([a + 2, 9])
J.gens()
J.gens_reduced()  # random sign
K.ideal([a + 2, 3]).gens_reduced()

gens_two()[source]¶

Express this ideal using exactly two generators, the first of which is a generator for the intersection of the ideal with \(\QQ\).

ALGORITHM: uses PARI’s pari:idealtwoelt function, which runs in randomized polynomial time and is very fast in practice.

EXAMPLES:

Sage

sage: x = polygen(ZZ)
sage: R.<x> = PolynomialRing(QQ)
sage: K.<a> = NumberField(x^2 + 5)
sage: J = K.ideal([a + 2, 9])
sage: J.gens()
(a + 2, 9)
sage: J.gens_two()
(9, a + 2)
sage: K.ideal([a + 5, a + 8]).gens_two()
(3, a + 2)
sage: K.ideal(0).gens_two()
(0, 0)

Python

>>> from sage.all import *
>>> x = polygen(ZZ)
>>> R = PolynomialRing(QQ, names=('x',)); (x,) = R._first_ngens(1)
>>> K = NumberField(x**Integer(2) + Integer(5), names=('a',)); (a,) = K._first_ngens(1)
>>> J = K.ideal([a + Integer(2), Integer(9)])
>>> J.gens()
(a + 2, 9)
>>> J.gens_two()
(9, a + 2)
>>> K.ideal([a + Integer(5), a + Integer(8)]).gens_two()
(3, a + 2)
>>> K.ideal(Integer(0)).gens_two()
(0, 0)

Sage Live

x = polygen(ZZ)
R.<x> = PolynomialRing(QQ)
K.<a> = NumberField(x^2 + 5)
J = K.ideal([a + 2, 9])
J.gens()
J.gens_two()
K.ideal([a + 5, a + 8]).gens_two()
K.ideal(0).gens_two()

The second generator is zero if and only if the ideal is generated by a rational, in contrast to the PARI function pari:idealtwoelt:

Sage

sage: I = K.ideal(12)
sage: pari(K).idealtwoelt(I)  # Note that second element is not zero
[12, [0, 12]~]
sage: I.gens_two()
(12, 0)

Python

>>> from sage.all import *
>>> I = K.ideal(Integer(12))
>>> pari(K).idealtwoelt(I)  # Note that second element is not zero
[12, [0, 12]~]
>>> I.gens_two()
(12, 0)

Sage Live

I = K.ideal(12)
pari(K).idealtwoelt(I)  # Note that second element is not zero
I.gens_two()

ideal_class_log(proof=None)[source]¶

Return the output of PARI’s pari:bnfisprincipal for this ideal, i.e. a vector expressing the class of this ideal in terms of a set of generators for the class group.

Since it uses the PARI method pari:bnfisprincipal, specify proof=True (this is the default setting) to prove the correctness of the output.

EXAMPLES:

When the class number is 1, the result is always the empty list:

Sage

sage: K.<a> = QuadraticField(-163)
sage: J = K.primes_above(random_prime(10^6))[0]
sage: J.ideal_class_log()
[]

Python

>>> from sage.all import *
>>> K = QuadraticField(-Integer(163), names=('a',)); (a,) = K._first_ngens(1)
>>> J = K.primes_above(random_prime(Integer(10)**Integer(6)))[Integer(0)]
>>> J.ideal_class_log()
[]

Sage Live

K.<a> = QuadraticField(-163)
J = K.primes_above(random_prime(10^6))[0]
J.ideal_class_log()

An example with class group of order 2. The first ideal is not principal, the second one is:

Sage

sage: K.<a> = QuadraticField(-5)
sage: J = K.ideal(23).factor()[0][0]
sage: J.ideal_class_log()
[1]
sage: (J^10).ideal_class_log()
[0]

Python

>>> from sage.all import *
>>> K = QuadraticField(-Integer(5), names=('a',)); (a,) = K._first_ngens(1)
>>> J = K.ideal(Integer(23)).factor()[Integer(0)][Integer(0)]
>>> J.ideal_class_log()
[1]
>>> (J**Integer(10)).ideal_class_log()
[0]

Sage Live

K.<a> = QuadraticField(-5)
J = K.ideal(23).factor()[0][0]
J.ideal_class_log()
(J^10).ideal_class_log()

An example with a more complicated class group:

Sage

sage: x = polygen(ZZ)
sage: K.<a, b> = NumberField([x^3 - x + 1, x^2 + 26])
sage: K.class_group()
Class group of order 18 with structure C6 x C3 of
 Number Field in a with defining polynomial x^3 - x + 1 over its base field
sage: K.primes_above(7)[0].ideal_class_log() # random
[1, 2]

Python

>>> from sage.all import *
>>> x = polygen(ZZ)
>>> K = NumberField([x**Integer(3) - x + Integer(1), x**Integer(2) + Integer(26)], names=('a', 'b',)); (a, b,) = K._first_ngens(2)
>>> K.class_group()
Class group of order 18 with structure C6 x C3 of
 Number Field in a with defining polynomial x^3 - x + 1 over its base field
>>> K.primes_above(Integer(7))[Integer(0)].ideal_class_log() # random
[1, 2]

Sage Live

x = polygen(ZZ)
K.<a, b> = NumberField([x^3 - x + 1, x^2 + 26])
K.class_group()
K.primes_above(7)[0].ideal_class_log() # random

inertia_group()[source]¶

Return the inertia group of self, i.e. the set of elements \(s\) of the Galois group of the number field of self (which we assume is Galois) such that \(s\) acts trivially modulo self. This is the same as the 0th ramification group of self. See the GaloisGroup_v2.inertia_group() method further examples and doctests.

EXAMPLES:

Sage

sage: QuadraticField(-23, 'w').primes_above(23)[0].inertia_group()          # needs sage.groups
Subgroup generated by [(1,2)] of (Galois group 2T1 (S2) with order 2 of x^2 + 23)

Python

>>> from sage.all import *
>>> QuadraticField(-Integer(23), 'w').primes_above(Integer(23))[Integer(0)].inertia_group()          # needs sage.groups
Subgroup generated by [(1,2)] of (Galois group 2T1 (S2) with order 2 of x^2 + 23)

Sage Live

QuadraticField(-23, 'w').primes_above(23)[0].inertia_group()          # needs sage.groups

integral_basis()[source]¶

Return a list of generators for this ideal as a \(\ZZ\)-module.

EXAMPLES:

Sage

sage: x = polygen(ZZ)
sage: R.<x> = PolynomialRing(QQ)
sage: K.<i> = NumberField(x^2 + 1)
sage: J = K.ideal(i + 1)
sage: J.integral_basis()
[2, i + 1]

Python

>>> from sage.all import *
>>> x = polygen(ZZ)
>>> R = PolynomialRing(QQ, names=('x',)); (x,) = R._first_ngens(1)
>>> K = NumberField(x**Integer(2) + Integer(1), names=('i',)); (i,) = K._first_ngens(1)
>>> J = K.ideal(i + Integer(1))
>>> J.integral_basis()
[2, i + 1]

Sage Live

x = polygen(ZZ)
R.<x> = PolynomialRing(QQ)
K.<i> = NumberField(x^2 + 1)
J = K.ideal(i + 1)
J.integral_basis()

integral_split()[source]¶

Return a tuple \((I, d)\), where \(I\) is an integral ideal, and \(d\) is the smallest positive integer such that this ideal is equal to \(I/d\).

EXAMPLES:

Sage

sage: x = polygen(ZZ)
sage: R.<x> = PolynomialRing(QQ)
sage: K.<a> = NumberField(x^2 - 5)
sage: I = K.ideal(2/(5+a))
sage: I.is_integral()
False
sage: J, d = I.integral_split()
sage: J
Fractional ideal (-1/2*a + 5/2)
sage: J.is_integral()
True
sage: d
5
sage: I == J/d
True

Python

>>> from sage.all import *
>>> x = polygen(ZZ)
>>> R = PolynomialRing(QQ, names=('x',)); (x,) = R._first_ngens(1)
>>> K = NumberField(x**Integer(2) - Integer(5), names=('a',)); (a,) = K._first_ngens(1)
>>> I = K.ideal(Integer(2)/(Integer(5)+a))
>>> I.is_integral()
False
>>> J, d = I.integral_split()
>>> J
Fractional ideal (-1/2*a + 5/2)
>>> J.is_integral()
True
>>> d
5
>>> I == J/d
True

Sage Live

x = polygen(ZZ)
R.<x> = PolynomialRing(QQ)
K.<a> = NumberField(x^2 - 5)
I = K.ideal(2/(5+a))
I.is_integral()
J, d = I.integral_split()
J
J.is_integral()
d
I == J/d

intersection(other)[source]¶

Return the intersection of self and other.

EXAMPLES:

Sage

sage: K.<a> = QuadraticField(-11)
sage: p = K.ideal((a + 1)/2); q = K.ideal((a + 3)/2)
sage: p.intersection(q) == q.intersection(p) == K.ideal(a - 2)
True

Python

>>> from sage.all import *
>>> K = QuadraticField(-Integer(11), names=('a',)); (a,) = K._first_ngens(1)
>>> p = K.ideal((a + Integer(1))/Integer(2)); q = K.ideal((a + Integer(3))/Integer(2))
>>> p.intersection(q) == q.intersection(p) == K.ideal(a - Integer(2))
True

Sage Live

K.<a> = QuadraticField(-11)
p = K.ideal((a + 1)/2); q = K.ideal((a + 3)/2)
p.intersection(q) == q.intersection(p) == K.ideal(a - 2)

An example with non-principal ideals:

Sage

sage: x = polygen(ZZ)
sage: L.<a> = NumberField(x^3 - 7)
sage: p = L.ideal(a^2 + a + 1, 2)
sage: q = L.ideal(a + 1)
sage: p.intersection(q) == L.ideal(8, 2*a + 2)
True

Python

>>> from sage.all import *
>>> x = polygen(ZZ)
>>> L = NumberField(x**Integer(3) - Integer(7), names=('a',)); (a,) = L._first_ngens(1)
>>> p = L.ideal(a**Integer(2) + a + Integer(1), Integer(2))
>>> q = L.ideal(a + Integer(1))
>>> p.intersection(q) == L.ideal(Integer(8), Integer(2)*a + Integer(2))
True

Sage Live

x = polygen(ZZ)
L.<a> = NumberField(x^3 - 7)
p = L.ideal(a^2 + a + 1, 2)
q = L.ideal(a + 1)
p.intersection(q) == L.ideal(8, 2*a + 2)

A relative example:

Sage

sage: L.<a,b> = NumberField([x^2 + 11, x^2 - 5])
sage: A = L.ideal([15, (-3/2*b + 7/2)*a - 8])
sage: B = L.ideal([6, (-1/2*b + 1)*a - b - 5/2])
sage: A.intersection(B) == L.ideal(-1/2*a - 3/2*b - 1)
True

Python

>>> from sage.all import *
>>> L = NumberField([x**Integer(2) + Integer(11), x**Integer(2) - Integer(5)], names=('a', 'b',)); (a, b,) = L._first_ngens(2)
>>> A = L.ideal([Integer(15), (-Integer(3)/Integer(2)*b + Integer(7)/Integer(2))*a - Integer(8)])
>>> B = L.ideal([Integer(6), (-Integer(1)/Integer(2)*b + Integer(1))*a - b - Integer(5)/Integer(2)])
>>> A.intersection(B) == L.ideal(-Integer(1)/Integer(2)*a - Integer(3)/Integer(2)*b - Integer(1))
True

Sage Live

L.<a,b> = NumberField([x^2 + 11, x^2 - 5])
A = L.ideal([15, (-3/2*b + 7/2)*a - 8])
B = L.ideal([6, (-1/2*b + 1)*a - b - 5/2])
A.intersection(B) == L.ideal(-1/2*a - 3/2*b - 1)

is_integral()[source]¶

Return True if this ideal is integral.

EXAMPLES:

Sage

sage: R.<x> = PolynomialRing(QQ)
sage: K.<a> = NumberField(x^5 - x + 1)
sage: K.ideal(a).is_integral()
True
sage: (K.ideal(1) / (3*a+1)).is_integral()
False

Python

>>> from sage.all import *
>>> R = PolynomialRing(QQ, names=('x',)); (x,) = R._first_ngens(1)
>>> K = NumberField(x**Integer(5) - x + Integer(1), names=('a',)); (a,) = K._first_ngens(1)
>>> K.ideal(a).is_integral()
True
>>> (K.ideal(Integer(1)) / (Integer(3)*a+Integer(1))).is_integral()
False

Sage Live

R.<x> = PolynomialRing(QQ)
K.<a> = NumberField(x^5 - x + 1)
K.ideal(a).is_integral()
(K.ideal(1) / (3*a+1)).is_integral()

is_maximal()[source]¶

Return True if this ideal is maximal. This is equivalent to self being prime and nonzero.

EXAMPLES:

Sage

sage: x = polygen(ZZ)
sage: K.<a> = NumberField(x^3 + 3); K
Number Field in a with defining polynomial x^3 + 3
sage: K.ideal(5).is_maximal()
False
sage: K.ideal(7).is_maximal()
True

Python

>>> from sage.all import *
>>> x = polygen(ZZ)
>>> K = NumberField(x**Integer(3) + Integer(3), names=('a',)); (a,) = K._first_ngens(1); K
Number Field in a with defining polynomial x^3 + 3
>>> K.ideal(Integer(5)).is_maximal()
False
>>> K.ideal(Integer(7)).is_maximal()
True

Sage Live

x = polygen(ZZ)
K.<a> = NumberField(x^3 + 3); K
K.ideal(5).is_maximal()
K.ideal(7).is_maximal()

is_prime()[source]¶

Return True if this ideal is prime.

EXAMPLES:

Sage

sage: x = polygen(ZZ)
sage: K.<a> = NumberField(x^2 - 17); K
Number Field in a with defining polynomial x^2 - 17
sage: K.ideal(5).is_prime()   # inert prime
True
sage: K.ideal(13).is_prime()  # split
False
sage: K.ideal(17).is_prime()  # ramified
False

Python

>>> from sage.all import *
>>> x = polygen(ZZ)
>>> K = NumberField(x**Integer(2) - Integer(17), names=('a',)); (a,) = K._first_ngens(1); K
Number Field in a with defining polynomial x^2 - 17
>>> K.ideal(Integer(5)).is_prime()   # inert prime
True
>>> K.ideal(Integer(13)).is_prime()  # split
False
>>> K.ideal(Integer(17)).is_prime()  # ramified
False

Sage Live

x = polygen(ZZ)
K.<a> = NumberField(x^2 - 17); K
K.ideal(5).is_prime()   # inert prime
K.ideal(13).is_prime()  # split
K.ideal(17).is_prime()  # ramified

is_principal(proof=None)[source]¶

Return True if this ideal is principal.

Since it uses the PARI method pari:bnfisprincipal, specify proof=True (this is the default setting) to prove the correctness of the output.

EXAMPLES:

Sage

sage: K = QuadraticField(-119,'a')
sage: P = K.factor(2)[1][0]
sage: P.is_principal()
False
sage: I = P^5
sage: I.is_principal()
True
sage: I  # random
Fractional ideal (-1/2*a + 3/2)
sage: P = K.ideal([2]).factor()[1][0]
sage: I = P^5
sage: I.is_principal()
True

Python

>>> from sage.all import *
>>> K = QuadraticField(-Integer(119),'a')
>>> P = K.factor(Integer(2))[Integer(1)][Integer(0)]
>>> P.is_principal()
False
>>> I = P**Integer(5)
>>> I.is_principal()
True
>>> I  # random
Fractional ideal (-1/2*a + 3/2)
>>> P = K.ideal([Integer(2)]).factor()[Integer(1)][Integer(0)]
>>> I = P**Integer(5)
>>> I.is_principal()
True

Sage Live

K = QuadraticField(-119,'a')
P = K.factor(2)[1][0]
P.is_principal()
I = P^5
I.is_principal()
I  # random
P = K.ideal([2]).factor()[1][0]
I = P^5
I.is_principal()

is_zero()[source]¶

Return True iff self is the zero ideal.

Note that \((0)\) is a NumberFieldIdeal, not a NumberFieldFractionalIdeal.

EXAMPLES:

Sage

sage: x = polygen(ZZ)
sage: K.<a> = NumberField(x^2 + 2); K
Number Field in a with defining polynomial x^2 + 2
sage: K.ideal(3).is_zero()
False
sage: I = K.ideal(0); I.is_zero()
True
sage: I
Ideal (0) of Number Field in a with defining polynomial x^2 + 2

Python

>>> from sage.all import *
>>> x = polygen(ZZ)
>>> K = NumberField(x**Integer(2) + Integer(2), names=('a',)); (a,) = K._first_ngens(1); K
Number Field in a with defining polynomial x^2 + 2
>>> K.ideal(Integer(3)).is_zero()
False
>>> I = K.ideal(Integer(0)); I.is_zero()
True
>>> I
Ideal (0) of Number Field in a with defining polynomial x^2 + 2

Sage Live

x = polygen(ZZ)
K.<a> = NumberField(x^2 + 2); K
K.ideal(3).is_zero()
I = K.ideal(0); I.is_zero()
I

norm()[source]¶

Return the norm of this fractional ideal as a rational number.

EXAMPLES:

Sage

sage: x = polygen(ZZ)
sage: K.<a> = NumberField(x^4 + 23); K
Number Field in a with defining polynomial x^4 + 23
sage: I = K.ideal(19); I
Fractional ideal (19)
sage: factor(I.norm())
19^4
sage: F = I.factor()
sage: F[0][0].norm().factor()
19^2

Python

>>> from sage.all import *
>>> x = polygen(ZZ)
>>> K = NumberField(x**Integer(4) + Integer(23), names=('a',)); (a,) = K._first_ngens(1); K
Number Field in a with defining polynomial x^4 + 23
>>> I = K.ideal(Integer(19)); I
Fractional ideal (19)
>>> factor(I.norm())
19^4
>>> F = I.factor()
>>> F[Integer(0)][Integer(0)].norm().factor()
19^2

Sage Live

x = polygen(ZZ)
K.<a> = NumberField(x^4 + 23); K
I = K.ideal(19); I
factor(I.norm())
F = I.factor()
F[0][0].norm().factor()

number_field()[source]¶

Return the number field that this is a fractional ideal in.

EXAMPLES:

Sage

sage: x = polygen(ZZ)
sage: K.<a> = NumberField(x^2 + 2); K
Number Field in a with defining polynomial x^2 + 2
sage: K.ideal(3).number_field()
Number Field in a with defining polynomial x^2 + 2
sage: K.ideal(0).number_field()  # not tested (not implemented)
Number Field in a with defining polynomial x^2 + 2

Python

>>> from sage.all import *
>>> x = polygen(ZZ)
>>> K = NumberField(x**Integer(2) + Integer(2), names=('a',)); (a,) = K._first_ngens(1); K
Number Field in a with defining polynomial x^2 + 2
>>> K.ideal(Integer(3)).number_field()
Number Field in a with defining polynomial x^2 + 2
>>> K.ideal(Integer(0)).number_field()  # not tested (not implemented)
Number Field in a with defining polynomial x^2 + 2

Sage Live

x = polygen(ZZ)
K.<a> = NumberField(x^2 + 2); K
K.ideal(3).number_field()
K.ideal(0).number_field()  # not tested (not implemented)

pari_hnf()[source]¶

Return PARI’s representation of this ideal in Hermite normal form.

EXAMPLES:

Sage

sage: x = polygen(ZZ)
sage: R.<x> = PolynomialRing(QQ)
sage: K.<a> = NumberField(x^3 - 2)
sage: I = K.ideal(2/(5+a))
sage: I.pari_hnf()
[2, 0, 50/127; 0, 2, 244/127; 0, 0, 2/127]

Python

>>> from sage.all import *
>>> x = polygen(ZZ)
>>> R = PolynomialRing(QQ, names=('x',)); (x,) = R._first_ngens(1)
>>> K = NumberField(x**Integer(3) - Integer(2), names=('a',)); (a,) = K._first_ngens(1)
>>> I = K.ideal(Integer(2)/(Integer(5)+a))
>>> I.pari_hnf()
[2, 0, 50/127; 0, 2, 244/127; 0, 0, 2/127]

Sage Live

x = polygen(ZZ)
R.<x> = PolynomialRing(QQ)
K.<a> = NumberField(x^3 - 2)
I = K.ideal(2/(5+a))
I.pari_hnf()

pari_prime()[source]¶

Return a PARI prime ideal corresponding to the ideal self.

INPUT:

self – a prime ideal

OUTPUT: a PARI “prime ideal”, i.e. a five-component vector \([p,a,e,f,b]\) representing the prime ideal \(p O_K + a O_K\), \(e\), \(f\) as usual, \(a\) as vector of components on the integral basis, \(b\) Lenstra’s constant.

EXAMPLES:

Sage

sage: K.<i> = QuadraticField(-1)
sage: K.ideal(3).pari_prime()
[3, [3, 0]~, 1, 2, 1]
sage: K.ideal(2+i).pari_prime()
[5, [2, 1]~, 1, 1, [-2, -1; 1, -2]]
sage: K.ideal(2).pari_prime()
Traceback (most recent call last):
...
ValueError: Fractional ideal (2) is not a prime ideal

Python

>>> from sage.all import *
>>> K = QuadraticField(-Integer(1), names=('i',)); (i,) = K._first_ngens(1)
>>> K.ideal(Integer(3)).pari_prime()
[3, [3, 0]~, 1, 2, 1]
>>> K.ideal(Integer(2)+i).pari_prime()
[5, [2, 1]~, 1, 1, [-2, -1; 1, -2]]
>>> K.ideal(Integer(2)).pari_prime()
Traceback (most recent call last):
...
ValueError: Fractional ideal (2) is not a prime ideal

Sage Live

K.<i> = QuadraticField(-1)
K.ideal(3).pari_prime()
K.ideal(2+i).pari_prime()
K.ideal(2).pari_prime()

ramification_group(v)[source]¶

Return the \(v\)-th ramification group of self, i.e. the set of elements \(s\) of the Galois group of the number field of self (which we assume is Galois) such that \(s\) acts trivially modulo the \((v+1)\)’st power of self. See the GaloisGroup.ramification_group() method for further examples and doctests.

EXAMPLES:

Sage

sage: QuadraticField(-23, 'w').primes_above(23)[0].ramification_group(0)    # needs sage.groups
Subgroup generated by [(1,2)] of (Galois group 2T1 (S2) with order 2 of x^2 + 23)
sage: QuadraticField(-23, 'w').primes_above(23)[0].ramification_group(1)    # needs sage.groups
Subgroup generated by [()] of (Galois group 2T1 (S2) with order 2 of x^2 + 23)

Python

>>> from sage.all import *
>>> QuadraticField(-Integer(23), 'w').primes_above(Integer(23))[Integer(0)].ramification_group(Integer(0))    # needs sage.groups
Subgroup generated by [(1,2)] of (Galois group 2T1 (S2) with order 2 of x^2 + 23)
>>> QuadraticField(-Integer(23), 'w').primes_above(Integer(23))[Integer(0)].ramification_group(Integer(1))    # needs sage.groups
Subgroup generated by [()] of (Galois group 2T1 (S2) with order 2 of x^2 + 23)

Sage Live

QuadraticField(-23, 'w').primes_above(23)[0].ramification_group(0)    # needs sage.groups
QuadraticField(-23, 'w').primes_above(23)[0].ramification_group(1)    # needs sage.groups

random_element(*args, **kwds)[source]¶

Return a random element of this ideal.

INPUT:

*args, *kwds – parameters passed to the random integer function. See the documentation of ZZ.random_element() for details.

OUTPUT:

A random element of this fractional ideal, computed as a random \(\ZZ\)-linear combination of the basis.

EXAMPLES:

Sage

sage: x = polygen(ZZ)
sage: K.<a> = NumberField(x^3 + 2)
sage: I = K.ideal(1 - a)
sage: I.random_element() # random output
-a^2 - a - 19
sage: I.random_element(distribution='uniform') # random output
a^2 - 2*a - 8
sage: I.random_element(-30, 30) # random output
-7*a^2 - 17*a - 75
sage: I.random_element(-100, 200).is_integral()
True
sage: I.random_element(-30, 30).parent() is K
True

Python

>>> from sage.all import *
>>> x = polygen(ZZ)
>>> K = NumberField(x**Integer(3) + Integer(2), names=('a',)); (a,) = K._first_ngens(1)
>>> I = K.ideal(Integer(1) - a)
>>> I.random_element() # random output
-a^2 - a - 19
>>> I.random_element(distribution='uniform') # random output
a^2 - 2*a - 8
>>> I.random_element(-Integer(30), Integer(30)) # random output
-7*a^2 - 17*a - 75
>>> I.random_element(-Integer(100), Integer(200)).is_integral()
True
>>> I.random_element(-Integer(30), Integer(30)).parent() is K
True

Sage Live

x = polygen(ZZ)
K.<a> = NumberField(x^3 + 2)
I = K.ideal(1 - a)
I.random_element() # random output
I.random_element(distribution='uniform') # random output
I.random_element(-30, 30) # random output
I.random_element(-100, 200).is_integral()
I.random_element(-30, 30).parent() is K

A relative example:

Sage

sage: K.<a, b> = NumberField([x^2 + 2, x^2 + 1000*x + 1])
sage: I = K.ideal(1 - a)
sage: I.random_element()  # random output
17/500002*a^3 + 737253/250001*a^2 - 1494505893/500002*a + 752473260/250001
sage: I.random_element().is_integral()
True
sage: I.random_element(-100, 200).parent() is K
True

Python

>>> from sage.all import *
>>> K = NumberField([x**Integer(2) + Integer(2), x**Integer(2) + Integer(1000)*x + Integer(1)], names=('a', 'b',)); (a, b,) = K._first_ngens(2)
>>> I = K.ideal(Integer(1) - a)
>>> I.random_element()  # random output
17/500002*a^3 + 737253/250001*a^2 - 1494505893/500002*a + 752473260/250001
>>> I.random_element().is_integral()
True
>>> I.random_element(-Integer(100), Integer(200)).parent() is K
True

Sage Live

K.<a, b> = NumberField([x^2 + 2, x^2 + 1000*x + 1])
I = K.ideal(1 - a)
I.random_element()  # random output
I.random_element().is_integral()
I.random_element(-100, 200).parent() is K

reduce_equiv()[source]¶

Return a small ideal that is equivalent to self in the group of fractional ideals modulo principal ideals. Very often (but not always) if self is principal then this function returns the unit ideal.

ALGORITHM: Calls pari:idealred function.

EXAMPLES:

Sage

sage: x = polygen(ZZ)
sage: K.<w> = NumberField(x^2 + 23)
sage: I = ideal(w*23^5); I
Fractional ideal (6436343*w)
sage: I.reduce_equiv()
Fractional ideal (1)
sage: I = K.class_group().0.ideal()^10; I
Fractional ideal (1024, 1/2*w + 979/2)
sage: I.reduce_equiv()
Fractional ideal (2, 1/2*w - 1/2)

Python

>>> from sage.all import *
>>> x = polygen(ZZ)
>>> K = NumberField(x**Integer(2) + Integer(23), names=('w',)); (w,) = K._first_ngens(1)
>>> I = ideal(w*Integer(23)**Integer(5)); I
Fractional ideal (6436343*w)
>>> I.reduce_equiv()
Fractional ideal (1)
>>> I = K.class_group().gen(0).ideal()**Integer(10); I
Fractional ideal (1024, 1/2*w + 979/2)
>>> I.reduce_equiv()
Fractional ideal (2, 1/2*w - 1/2)

Sage Live

x = polygen(ZZ)
K.<w> = NumberField(x^2 + 23)
I = ideal(w*23^5); I
I.reduce_equiv()
I = K.class_group().0.ideal()^10; I
I.reduce_equiv()

relative_norm()[source]¶

A synonym for norm().

EXAMPLES:

Sage

sage: x = polygen(ZZ)
sage: K.<i> = NumberField(x^2 + 1)
sage: K.ideal(1 + 2*i).relative_norm()
5

Python

>>> from sage.all import *
>>> x = polygen(ZZ)
>>> K = NumberField(x**Integer(2) + Integer(1), names=('i',)); (i,) = K._first_ngens(1)
>>> K.ideal(Integer(1) + Integer(2)*i).relative_norm()
5

Sage Live

x = polygen(ZZ)
K.<i> = NumberField(x^2 + 1)
K.ideal(1 + 2*i).relative_norm()

relative_ramification_index()[source]¶

A synonym for ramification_index().

EXAMPLES:

Sage

sage: x = polygen(ZZ)
sage: K.<i> = NumberField(x^2 + 1)
sage: K.ideal(1 + i).relative_ramification_index()
2

Python

>>> from sage.all import *
>>> x = polygen(ZZ)
>>> K = NumberField(x**Integer(2) + Integer(1), names=('i',)); (i,) = K._first_ngens(1)
>>> K.ideal(Integer(1) + i).relative_ramification_index()
2

Sage Live

x = polygen(ZZ)
K.<i> = NumberField(x^2 + 1)
K.ideal(1 + i).relative_ramification_index()

residue_symbol(e, m, check=True)[source]¶

The \(m\)-th power residue symbol for an element \(e\) and the proper ideal.

\[\left(\frac{\alpha}{\mathbf{P}}\right) \equiv \alpha^{\frac{N(\mathbf{P})-1}{m}} \operatorname{mod} \mathbf{P}\]

Note

accepts \(m=1\), in which case returns 1

Note

can also be called for an element from sage.rings.number_field_element.residue_symbol

Note

\(e\) is coerced into the number field of self

Note

if \(m=2\), \(e\) is an integer, and self.number_field() has absolute degree 1 (i.e. it is a copy of the rationals), then this calls kronecker_symbol(), which is implemented using GMP.

INPUT:

e – element of the number field
m – positive integer

OUTPUT: an \(m\)-th root of unity in the number field

EXAMPLES:

Quadratic Residue (7 is not a square modulo 11):

Sage

sage: x = polygen(ZZ)
sage: K.<a> = NumberField(x - 1)
sage: K.ideal(11).residue_symbol(7,2)
-1

Python

>>> from sage.all import *
>>> x = polygen(ZZ)
>>> K = NumberField(x - Integer(1), names=('a',)); (a,) = K._first_ngens(1)
>>> K.ideal(Integer(11)).residue_symbol(Integer(7),Integer(2))
-1

Sage Live

x = polygen(ZZ)
K.<a> = NumberField(x - 1)
K.ideal(11).residue_symbol(7,2)

Cubic Residue:

Sage

sage: K.<w> = NumberField(x^2 - x + 1)
sage: K.ideal(17).residue_symbol(w^2 + 3, 3)
-w

Python

>>> from sage.all import *
>>> K = NumberField(x**Integer(2) - x + Integer(1), names=('w',)); (w,) = K._first_ngens(1)
>>> K.ideal(Integer(17)).residue_symbol(w**Integer(2) + Integer(3), Integer(3))
-w

Sage Live

K.<w> = NumberField(x^2 - x + 1)
K.ideal(17).residue_symbol(w^2 + 3, 3)

The field must contain the \(m\)-th roots of unity:

Sage

sage: K.<w> = NumberField(x^2 - x + 1)
sage: K.ideal(17).residue_symbol(w^2 + 3, 5)
Traceback (most recent call last):
...
ValueError: The residue symbol to that power is not defined for the number field

Python

>>> from sage.all import *
>>> K = NumberField(x**Integer(2) - x + Integer(1), names=('w',)); (w,) = K._first_ngens(1)
>>> K.ideal(Integer(17)).residue_symbol(w**Integer(2) + Integer(3), Integer(5))
Traceback (most recent call last):
...
ValueError: The residue symbol to that power is not defined for the number field

Sage Live

K.<w> = NumberField(x^2 - x + 1)
K.ideal(17).residue_symbol(w^2 + 3, 5)

smallest_integer()[source]¶

Return the smallest nonnegative integer in \(I \cap \ZZ\), where \(I\) is this ideal. If \(I = 0\), returns 0.

EXAMPLES:

Sage

sage: R.<x> = PolynomialRing(QQ)
sage: K.<a> = NumberField(x^2 + 6)
sage: I = K.ideal([4, a])/7; I
Fractional ideal (2/7, 1/7*a)
sage: I.smallest_integer()
2

Python

>>> from sage.all import *
>>> R = PolynomialRing(QQ, names=('x',)); (x,) = R._first_ngens(1)
>>> K = NumberField(x**Integer(2) + Integer(6), names=('a',)); (a,) = K._first_ngens(1)
>>> I = K.ideal([Integer(4), a])/Integer(7); I
Fractional ideal (2/7, 1/7*a)
>>> I.smallest_integer()
2

Sage Live

R.<x> = PolynomialRing(QQ)
K.<a> = NumberField(x^2 + 6)
I = K.ideal([4, a])/7; I
I.smallest_integer()

valuation(p)[source]¶

Return the valuation of self at p.

INPUT:

p – a prime ideal \(\mathfrak{p}\) of this number field

OUTPUT:

(integer) The valuation of this fractional ideal at the prime \(\mathfrak{p}\). If \(\mathfrak{p}\) is not prime, raise a ValueError.

EXAMPLES:

Sage

sage: x = polygen(ZZ)
sage: K.<a> = NumberField(x^5 + 2); K
Number Field in a with defining polynomial x^5 + 2
sage: i = K.ideal(38); i
Fractional ideal (38)
sage: i.valuation(K.factor(19)[0][0])
1
sage: i.valuation(K.factor(2)[0][0])
5
sage: i.valuation(K.factor(3)[0][0])
0
sage: i.valuation(0)
Traceback (most recent call last):
...
ValueError: p (= Ideal (0) of Number Field in a
with defining polynomial x^5 + 2) must be nonzero
sage: K.ideal(0).valuation(K.factor(2)[0][0])
+Infinity

Python

>>> from sage.all import *
>>> x = polygen(ZZ)
>>> K = NumberField(x**Integer(5) + Integer(2), names=('a',)); (a,) = K._first_ngens(1); K
Number Field in a with defining polynomial x^5 + 2
>>> i = K.ideal(Integer(38)); i
Fractional ideal (38)
>>> i.valuation(K.factor(Integer(19))[Integer(0)][Integer(0)])
1
>>> i.valuation(K.factor(Integer(2))[Integer(0)][Integer(0)])
5
>>> i.valuation(K.factor(Integer(3))[Integer(0)][Integer(0)])
0
>>> i.valuation(Integer(0))
Traceback (most recent call last):
...
ValueError: p (= Ideal (0) of Number Field in a
with defining polynomial x^5 + 2) must be nonzero
>>> K.ideal(Integer(0)).valuation(K.factor(Integer(2))[Integer(0)][Integer(0)])
+Infinity

Sage Live

x = polygen(ZZ)
K.<a> = NumberField(x^5 + 2); K
i = K.ideal(38); i
i.valuation(K.factor(19)[0][0])
i.valuation(K.factor(2)[0][0])
i.valuation(K.factor(3)[0][0])
i.valuation(0)
K.ideal(0).valuation(K.factor(2)[0][0])

class sage.rings.number_field.number_field_ideal.QuotientMap(K, M_OK_change, Q, I)[source]¶

Bases: object

Class to hold data needed by quotient maps from number field orders to residue fields. These are only partial maps: the exact domain is the appropriate valuation ring. For examples, see residue_field().

sage.rings.number_field.number_field_ideal.basis_to_module(B, K)[source]¶

Given a basis \(B\) of elements for a \(\ZZ\)-submodule of a number field \(K\), return the corresponding \(\ZZ\)-submodule.

EXAMPLES:

Sage

sage: x = polygen(ZZ)
sage: K.<w> = NumberField(x^4 + 1)
sage: from sage.rings.number_field.number_field_ideal import basis_to_module
sage: basis_to_module([K.0, K.0^2 + 3], K)
Free module of degree 4 and rank 2 over Integer Ring
User basis matrix:
[0 1 0 0]
[3 0 1 0]

Python

>>> from sage.all import *
>>> x = polygen(ZZ)
>>> K = NumberField(x**Integer(4) + Integer(1), names=('w',)); (w,) = K._first_ngens(1)
>>> from sage.rings.number_field.number_field_ideal import basis_to_module
>>> basis_to_module([K.gen(0), K.gen(0)**Integer(2) + Integer(3)], K)
Free module of degree 4 and rank 2 over Integer Ring
User basis matrix:
[0 1 0 0]
[3 0 1 0]

Sage Live

x = polygen(ZZ)
K.<w> = NumberField(x^4 + 1)
from sage.rings.number_field.number_field_ideal import basis_to_module
basis_to_module([K.0, K.0^2 + 3], K)

sage.rings.number_field.number_field_ideal.is_NumberFieldFractionalIdeal(x)[source]¶

Return True if \(x\) is a fractional ideal of a number field.

EXAMPLES:

Sage

sage: from sage.rings.number_field.number_field_ideal import is_NumberFieldFractionalIdeal
sage: is_NumberFieldFractionalIdeal(2/3)
doctest:warning...
DeprecationWarning: The function is_NumberFieldFractionalIdeal is deprecated;
use 'isinstance(..., NumberFieldFractionalIdeal)' instead.
See https://github.com/sagemath/sage/issues/38124 for details.
False
sage: is_NumberFieldFractionalIdeal(ideal(5))
False
sage: x = polygen(ZZ)
sage: k.<a> = NumberField(x^2 + 2)
sage: I = k.ideal([a + 1]); I
Fractional ideal (a + 1)
sage: is_NumberFieldFractionalIdeal(I)
True
sage: Z = k.ideal(0); Z
Ideal (0) of Number Field in a with defining polynomial x^2 + 2
sage: is_NumberFieldFractionalIdeal(Z)
False

Python

>>> from sage.all import *
>>> from sage.rings.number_field.number_field_ideal import is_NumberFieldFractionalIdeal
>>> is_NumberFieldFractionalIdeal(Integer(2)/Integer(3))
doctest:warning...
DeprecationWarning: The function is_NumberFieldFractionalIdeal is deprecated;
use 'isinstance(..., NumberFieldFractionalIdeal)' instead.
See https://github.com/sagemath/sage/issues/38124 for details.
False
>>> is_NumberFieldFractionalIdeal(ideal(Integer(5)))
False
>>> x = polygen(ZZ)
>>> k = NumberField(x**Integer(2) + Integer(2), names=('a',)); (a,) = k._first_ngens(1)
>>> I = k.ideal([a + Integer(1)]); I
Fractional ideal (a + 1)
>>> is_NumberFieldFractionalIdeal(I)
True
>>> Z = k.ideal(Integer(0)); Z
Ideal (0) of Number Field in a with defining polynomial x^2 + 2
>>> is_NumberFieldFractionalIdeal(Z)
False

Sage Live

from sage.rings.number_field.number_field_ideal import is_NumberFieldFractionalIdeal
is_NumberFieldFractionalIdeal(2/3)
is_NumberFieldFractionalIdeal(ideal(5))
x = polygen(ZZ)
k.<a> = NumberField(x^2 + 2)
I = k.ideal([a + 1]); I
is_NumberFieldFractionalIdeal(I)
Z = k.ideal(0); Z
is_NumberFieldFractionalIdeal(Z)

sage.rings.number_field.number_field_ideal.is_NumberFieldIdeal(x)[source]¶

Return True if \(x\) is an ideal of a number field.

EXAMPLES:

Sage

sage: from sage.rings.number_field.number_field_ideal import is_NumberFieldIdeal
sage: is_NumberFieldIdeal(2/3)
doctest:warning...
DeprecationWarning: The function is_NumberFieldIdeal is deprecated;
use 'isinstance(..., NumberFieldIdeal)' instead.
See https://github.com/sagemath/sage/issues/38124 for details.
False
sage: is_NumberFieldIdeal(ideal(5))
False

sage: x = polygen(ZZ)
sage: k.<a> = NumberField(x^2 + 2)
sage: I = k.ideal([a + 1]); I
Fractional ideal (a + 1)
sage: is_NumberFieldIdeal(I)
True
sage: Z = k.ideal(0); Z
Ideal (0) of Number Field in a with defining polynomial x^2 + 2
sage: is_NumberFieldIdeal(Z)
True

Python

>>> from sage.all import *
>>> from sage.rings.number_field.number_field_ideal import is_NumberFieldIdeal
>>> is_NumberFieldIdeal(Integer(2)/Integer(3))
doctest:warning...
DeprecationWarning: The function is_NumberFieldIdeal is deprecated;
use 'isinstance(..., NumberFieldIdeal)' instead.
See https://github.com/sagemath/sage/issues/38124 for details.
False
>>> is_NumberFieldIdeal(ideal(Integer(5)))
False

>>> x = polygen(ZZ)
>>> k = NumberField(x**Integer(2) + Integer(2), names=('a',)); (a,) = k._first_ngens(1)
>>> I = k.ideal([a + Integer(1)]); I
Fractional ideal (a + 1)
>>> is_NumberFieldIdeal(I)
True
>>> Z = k.ideal(Integer(0)); Z
Ideal (0) of Number Field in a with defining polynomial x^2 + 2
>>> is_NumberFieldIdeal(Z)
True

Sage Live

from sage.rings.number_field.number_field_ideal import is_NumberFieldIdeal
is_NumberFieldIdeal(2/3)
is_NumberFieldIdeal(ideal(5))
x = polygen(ZZ)
k.<a> = NumberField(x^2 + 2)
I = k.ideal([a + 1]); I
is_NumberFieldIdeal(I)
Z = k.ideal(0); Z
is_NumberFieldIdeal(Z)

sage.rings.number_field.number_field_ideal.quotient_char_p(I, p)[source]¶

Given an integral ideal \(I\) that contains a prime number \(p\), compute a vector space \(V = (O_K \mod p) / (I \mod p)\), along with a homomorphism \(O_K \to V\) and a section \(V \to O_K\).

EXAMPLES:

Sage

sage: from sage.rings.number_field.number_field_ideal import quotient_char_p

sage: x = polygen(ZZ)
sage: K.<i> = NumberField(x^2 + 1); O = K.maximal_order(); I = K.fractional_ideal(15)
sage: quotient_char_p(I, 5)[0]
Vector space quotient V/W of dimension 2 over Finite Field of size 5 where
V: Vector space of dimension 2 over Finite Field of size 5
W: Vector space of degree 2 and dimension 0 over Finite Field of size 5
Basis matrix:
[]
sage: quotient_char_p(I, 3)[0]
Vector space quotient V/W of dimension 2 over Finite Field of size 3 where
V: Vector space of dimension 2 over Finite Field of size 3
W: Vector space of degree 2 and dimension 0 over Finite Field of size 3
Basis matrix:
[]

sage: I = K.factor(13)[0][0]; I
Fractional ideal (-2*i + 3)
sage: I.residue_class_degree()
1
sage: quotient_char_p(I, 13)[0]
Vector space quotient V/W of dimension 1 over Finite Field of size 13 where
V: Vector space of dimension 2 over Finite Field of size 13
W: Vector space of degree 2 and dimension 1 over Finite Field of size 13
Basis matrix:
[1 8]

Python

>>> from sage.all import *
>>> from sage.rings.number_field.number_field_ideal import quotient_char_p

>>> x = polygen(ZZ)
>>> K = NumberField(x**Integer(2) + Integer(1), names=('i',)); (i,) = K._first_ngens(1); O = K.maximal_order(); I = K.fractional_ideal(Integer(15))
>>> quotient_char_p(I, Integer(5))[Integer(0)]
Vector space quotient V/W of dimension 2 over Finite Field of size 5 where
V: Vector space of dimension 2 over Finite Field of size 5
W: Vector space of degree 2 and dimension 0 over Finite Field of size 5
Basis matrix:
[]
>>> quotient_char_p(I, Integer(3))[Integer(0)]
Vector space quotient V/W of dimension 2 over Finite Field of size 3 where
V: Vector space of dimension 2 over Finite Field of size 3
W: Vector space of degree 2 and dimension 0 over Finite Field of size 3
Basis matrix:
[]

>>> I = K.factor(Integer(13))[Integer(0)][Integer(0)]; I
Fractional ideal (-2*i + 3)
>>> I.residue_class_degree()
1
>>> quotient_char_p(I, Integer(13))[Integer(0)]
Vector space quotient V/W of dimension 1 over Finite Field of size 13 where
V: Vector space of dimension 2 over Finite Field of size 13
W: Vector space of degree 2 and dimension 1 over Finite Field of size 13
Basis matrix:
[1 8]

Sage Live

from sage.rings.number_field.number_field_ideal import quotient_char_p
x = polygen(ZZ)
K.<i> = NumberField(x^2 + 1); O = K.maximal_order(); I = K.fractional_ideal(15)
quotient_char_p(I, 5)[0]
quotient_char_p(I, 3)[0]
I = K.factor(13)[0][0]; I
I.residue_class_degree()
quotient_char_p(I, 13)[0]