Relative Number Field Ideals#

AUTHORS:

Steven Sivek (2005-05-16)
William Stein (2007-09-06)
Nick Alexander (2009-01)

EXAMPLES:

sage: x = polygen(ZZ, 'x')
sage: K.<a,b> = NumberField([x^2 + 1, x^2 + 2])
sage: A = K.absolute_field('z')
sage: I = A.factor(7)[0][0]
sage: from_A, to_A = A.structure()
sage: G = [from_A(z) for z in I.gens()]; G
[7, -2*b*a - 1]
sage: K.fractional_ideal(G)
Fractional ideal ((1/2*b + 2)*a - 1/2*b + 2)
sage: K.fractional_ideal(G).absolute_norm().factor()
7^2

x = polygen(ZZ, 'x')
K.<a,b> = NumberField([x^2 + 1, x^2 + 2])
A = K.absolute_field('z')
I = A.factor(7)[0][0]
from_A, to_A = A.structure()
G = [from_A(z) for z in I.gens()]; G
K.fractional_ideal(G)
K.fractional_ideal(G).absolute_norm().factor()

class sage.rings.number_field.number_field_ideal_rel.NumberFieldFractionalIdeal_rel(field, gens, coerce=True)#

Bases: NumberFieldFractionalIdeal

An ideal of a relative number field.

EXAMPLES:

sage: x = polygen(ZZ, 'x')
sage: K.<a> = NumberField([x^2 + 1, x^2 + 2]); K
Number Field in a0 with defining polynomial x^2 + 1 over its base field
sage: i = K.ideal(38); i
Fractional ideal (38)

sage: K.<a0, a1> = NumberField([x^2 + 1, x^2 + 2]); K
Number Field in a0 with defining polynomial x^2 + 1 over its base field
sage: i = K.ideal([a0+1]); i # random
Fractional ideal (-a1*a0)
sage: (g, ) = i.gens_reduced(); g # random
-a1*a0
sage: (g / (a0 + 1)).is_integral()
True
sage: ((a0 + 1) / g).is_integral()
True

x = polygen(ZZ, 'x')
K.<a> = NumberField([x^2 + 1, x^2 + 2]); K
i = K.ideal(38); i
K.<a0, a1> = NumberField([x^2 + 1, x^2 + 2]); K
i = K.ideal([a0+1]); i # random
(g, ) = i.gens_reduced(); g # random
(g / (a0 + 1)).is_integral()
((a0 + 1) / g).is_integral()

absolute_ideal(names='a')#

If this is an ideal in the extension $L / K$ , return the ideal with the same generators in the absolute field $L / Q$ .

INPUT:

names (optional) – string; name of generator of the absolute field

EXAMPLES:

sage: x = ZZ['x'].0
sage: K.<b> = NumberField(x^2 - 2)
sage: L.<c> = K.extension(x^2 - b)
sage: F.<m> = L.absolute_field()

x = ZZ['x'].0
K.<b> = NumberField(x^2 - 2)
L.<c> = K.extension(x^2 - b)
F.<m> = L.absolute_field()

An example of an inert ideal:

sage: P = F.factor(13)[0][0]; P
Fractional ideal (13)
sage: J = L.ideal(13)
sage: J.absolute_ideal()
Fractional ideal (13)

P = F.factor(13)[0][0]; P
J = L.ideal(13)
J.absolute_ideal()

Now a non-trivial ideal in $L$ that is principal in the subfield $K$ . Since the optional names argument is not passed, the generators of the absolute ideal $J$ are returned in terms of the default field generator $a$ . This does not agree with the generator $m$ of the absolute field $F$ defined above:

sage: J = L.ideal(b); J
Fractional ideal (b)
sage: J.absolute_ideal()
Fractional ideal (a^2)
sage: J.relative_norm()
Fractional ideal (2)
sage: J.absolute_norm()
4
sage: J.absolute_ideal().norm()
4

J = L.ideal(b); J
J.absolute_ideal()
J.relative_norm()
J.absolute_norm()
J.absolute_ideal().norm()

Now pass $m$ as the name for the generator of the absolute field:

sage: J.absolute_ideal('m')
Fractional ideal (m^2)

J.absolute_ideal('m')

Now an ideal not generated by an element of $K$ :

sage: J = L.ideal(c); J
Fractional ideal (c)
sage: J.absolute_ideal()
Fractional ideal (a)
sage: J.absolute_norm()
2
sage: J.ideal_below()
Fractional ideal (b)
sage: J.ideal_below().norm()
2

J = L.ideal(c); J
J.absolute_ideal()
J.absolute_norm()
J.ideal_below()
J.ideal_below().norm()

absolute_norm()#

Compute the absolute norm of this fractional ideal in a relative number field, returning a positive integer.

EXAMPLES:

sage: x = polygen(ZZ, 'x')
sage: L.<a, b, c> = QQ.extension([x^2 - 23, x^2 - 5, x^2 - 7])
sage: I = L.ideal(a + b)
sage: I.absolute_norm()
104976
sage: I.relative_norm().relative_norm().relative_norm()
104976

x = polygen(ZZ, 'x')
L.<a, b, c> = QQ.extension([x^2 - 23, x^2 - 5, x^2 - 7])
I = L.ideal(a + b)
I.absolute_norm()
I.relative_norm().relative_norm().relative_norm()

absolute_ramification_index()#

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

The absolute ramification index is the power of this prime appearing in the factorization of the rational prime that this prime lies over.

Use relative_ramification_index() to obtain the power of this prime occurring in the factorization of the prime ideal of the base field that this prime lies over.

EXAMPLES:

sage: PQ.<X> = QQ[]
sage: F.<a, b> = NumberFieldTower([X^2 - 2, X^2 - 3])
sage: PF.<Y> = F[]
sage: K.<c> = F.extension(Y^2 - (1 + a)*(a + b)*a*b)
sage: I = K.ideal(3, c)
sage: I.absolute_ramification_index()
4
sage: I.smallest_integer()
3
sage: K.ideal(3) == I^4
True

PQ.<X> = QQ[]
F.<a, b> = NumberFieldTower([X^2 - 2, X^2 - 3])
PF.<Y> = F[]
K.<c> = F.extension(Y^2 - (1 + a)*(a + b)*a*b)
I = K.ideal(3, c)
I.absolute_ramification_index()
I.smallest_integer()
K.ideal(3) == I^4

element_1_mod(other)#

Returns 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.

EXAMPLES:

sage: x = polygen(ZZ, 'x')
sage: K.<a, b> = NumberFieldTower([x^2 - 23, x^2 + 1])
sage: I = Ideal(2, (a - 3*b + 2)/2)
sage: J = K.ideal(a)
sage: z = I.element_1_mod(J)
sage: z in I
True
sage: 1 - z in J
True

x = polygen(ZZ, 'x')
K.<a, b> = NumberFieldTower([x^2 - 23, x^2 + 1])
I = Ideal(2, (a - 3*b + 2)/2)
J = K.ideal(a)
z = I.element_1_mod(J)
z in I
1 - z in J

factor()#

Factor the ideal by factoring the corresponding ideal in the absolute number field.

EXAMPLES:

sage: x = polygen(ZZ, 'x')
sage: K.<a, b> = QQ.extension([x^2 + 11, x^2 - 5])
sage: K.factor(5)
(Fractional ideal (5, (-1/4*b - 1/4)*a + 1/4*b - 3/4))^2
 * (Fractional ideal (5, (-1/4*b - 1/4)*a + 1/4*b - 7/4))^2
sage: K.ideal(5).factor()
(Fractional ideal (5, (-1/4*b - 1/4)*a + 1/4*b - 3/4))^2
 * (Fractional ideal (5, (-1/4*b - 1/4)*a + 1/4*b - 7/4))^2
sage: K.ideal(5).prime_factors()
[Fractional ideal (5, (-1/4*b - 1/4)*a + 1/4*b - 3/4),
 Fractional ideal (5, (-1/4*b - 1/4)*a + 1/4*b - 7/4)]

sage: PQ.<X> = QQ[]
sage: F.<a, b> = NumberFieldTower([X^2 - 2, X^2 - 3])
sage: PF.<Y> = F[]
sage: K.<c> = F.extension(Y^2 - (1 + a)*(a + b)*a*b)
sage: I = K.ideal(c)
sage: P = K.ideal((b*a - b - 1)*c/2 + a - 1)
sage: Q = K.ideal((b*a - b - 1)*c/2)
sage: list(I.factor()) == [(P, 2), (Q, 1)]
True
sage: I == P^2*Q
True
sage: [p.is_prime() for p in [P, Q]]
[True, True]

x = polygen(ZZ, 'x')
K.<a, b> = QQ.extension([x^2 + 11, x^2 - 5])
K.factor(5)
K.ideal(5).factor()
K.ideal(5).prime_factors()
PQ.<X> = QQ[]
F.<a, b> = NumberFieldTower([X^2 - 2, X^2 - 3])
PF.<Y> = F[]
K.<c> = F.extension(Y^2 - (1 + a)*(a + b)*a*b)
I = K.ideal(c)
P = K.ideal((b*a - b - 1)*c/2 + a - 1)
Q = K.ideal((b*a - b - 1)*c/2)
list(I.factor()) == [(P, 2), (Q, 1)]
I == P^2*Q
[p.is_prime() for p in [P, Q]]

free_module()#

Return this ideal as a $Z$ -submodule of the $Q$ -vector space corresponding to the ambient number field.

EXAMPLES:

sage: x = polygen(ZZ, 'x')
sage: K.<a, b> = NumberField([x^3 - x + 1, x^2 + 23])
sage: I = K.ideal(a*b - 1)
sage: I.free_module()
Free module of degree 6 and rank 6 over Integer Ring
User basis matrix:
...
sage: I.free_module().is_submodule(K.maximal_order().free_module())
True

x = polygen(ZZ, 'x')
K.<a, b> = NumberField([x^3 - x + 1, x^2 + 23])
I = K.ideal(a*b - 1)
I.free_module()
I.free_module().is_submodule(K.maximal_order().free_module())

gens_reduced()#

Return a small set of generators for this ideal. This will always return a single generator if one exists (i.e. if the ideal is principal), and otherwise two generators.

EXAMPLES:

sage: x = polygen(ZZ, 'x')
sage: K.<a, b> = NumberField([x^2 + 1, x^2 - 2])
sage: I = K.ideal((a + 1)*b/2 + 1)
sage: I.gens_reduced()
(1/2*b*a + 1/2*b + 1,)

x = polygen(ZZ, 'x')
K.<a, b> = NumberField([x^2 + 1, x^2 - 2])
I = K.ideal((a + 1)*b/2 + 1)
I.gens_reduced()

ideal_below()#

Compute the ideal of $K$ below this ideal of $L$ .

EXAMPLES:

sage: R.<x> = QQ[]
sage: K.<a> = NumberField(x^2 + 6)
sage: L.<b> = K.extension(K['x'].gen()^4 + a)
sage: N = L.ideal(b)
sage: M = N.ideal_below(); M == K.ideal([-a])
True
sage: Np = L.ideal([L(t) for t in M.gens()])
sage: Np.ideal_below() == M
True
sage: M.parent()
Monoid of ideals of Number Field in a with defining polynomial x^2 + 6
sage: M.ring()
Number Field in a with defining polynomial x^2 + 6
sage: M.ring() is K
True

R.<x> = QQ[]
K.<a> = NumberField(x^2 + 6)
L.<b> = K.extension(K['x'].gen()^4 + a)
N = L.ideal(b)
M = N.ideal_below(); M == K.ideal([-a])
Np = L.ideal([L(t) for t in M.gens()])
Np.ideal_below() == M
M.parent()
M.ring()
M.ring() is K

This example concerns an inert ideal:

sage: K = NumberField(x^4 + 6*x^2 + 24, 'a')
sage: K.factor(7)
Fractional ideal (7)
sage: K0, K0_into_K, _ = K.subfields(2)[0]
sage: K0
Number Field in a0 with defining polynomial x^2 - 6*x + 24
sage: L = K.relativize(K0_into_K, 'c'); L
Number Field in c with defining polynomial x^2 + a0 over its base field
sage: L.base_field() is K0
True
sage: L.ideal(7)
Fractional ideal (7)
sage: L.ideal(7).ideal_below()
Fractional ideal (7)
sage: L.ideal(7).ideal_below().number_field() is K0
True

K = NumberField(x^4 + 6*x^2 + 24, 'a')
K.factor(7)
K0, K0_into_K, _ = K.subfields(2)[0]
K0
L = K.relativize(K0_into_K, 'c'); L
L.base_field() is K0
L.ideal(7)
L.ideal(7).ideal_below()
L.ideal(7).ideal_below().number_field() is K0

This example concerns an ideal that splits in the quadratic field but each factor ideal remains inert in the extension:

sage: len(K.factor(19))
2
sage: K0 = L.base_field(); a0 = K0.gen()
sage: len(K0.factor(19))
2
sage: w1 = -a0 + 1; P1 = K0.ideal([w1])
sage: P1.norm().factor(), P1.is_prime()
(19, True)
sage: L_into_K, K_into_L = L.structure()
sage: L.ideal(K_into_L(K0_into_K(w1))).ideal_below() == P1
True

len(K.factor(19))
K0 = L.base_field(); a0 = K0.gen()
len(K0.factor(19))
w1 = -a0 + 1; P1 = K0.ideal([w1])
P1.norm().factor(), P1.is_prime()
L_into_K, K_into_L = L.structure()
L.ideal(K_into_L(K0_into_K(w1))).ideal_below() == P1

The choice of embedding of quadratic field into quartic field matters:

sage: rho, tau = K0.embeddings(K)
sage: L1 = K.relativize(rho, 'b')
sage: L2 = K.relativize(tau, 'b')
sage: L1_into_K, K_into_L1 = L1.structure()
sage: L2_into_K, K_into_L2 = L2.structure()
sage: a = K.gen()
sage: P = K.ideal([a^2 + 5])
sage: K_into_L1(P).ideal_below() == K0.ideal([-a0 + 1])
True
sage: K_into_L2(P).ideal_below() == K0.ideal([-a0 + 5])
True
sage: K0.ideal([-a0 + 1]) == K0.ideal([-a0 + 5])
False

rho, tau = K0.embeddings(K)
L1 = K.relativize(rho, 'b')
L2 = K.relativize(tau, 'b')
L1_into_K, K_into_L1 = L1.structure()
L2_into_K, K_into_L2 = L2.structure()
a = K.gen()
P = K.ideal([a^2 + 5])
K_into_L1(P).ideal_below() == K0.ideal([-a0 + 1])
K_into_L2(P).ideal_below() == K0.ideal([-a0 + 5])
K0.ideal([-a0 + 1]) == K0.ideal([-a0 + 5])

It works when the base field is itself a relative number field:

sage: PQ.<X> = QQ[]
sage: F.<a, b> = NumberFieldTower([X^2 - 2, X^2 - 3])
sage: PF.<Y> = F[]
sage: K.<c> = F.extension(Y^2 - (1 + a)*(a + b)*a*b)
sage: I = K.ideal(3, c)
sage: J = I.ideal_below()
sage: J == K.ideal(b)
True
sage: J.number_field() == F
True

PQ.<X> = QQ[]
F.<a, b> = NumberFieldTower([X^2 - 2, X^2 - 3])
PF.<Y> = F[]
K.<c> = F.extension(Y^2 - (1 + a)*(a + b)*a*b)
I = K.ideal(3, c)
J = I.ideal_below()
J == K.ideal(b)
J.number_field() == F

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

sage: K.<a> = NumberField(2*x^2 - 1/3)
sage: L.<b> = K.extension(5*x^2 + 1)
sage: P = L.primes_above(2)[0]
sage: P.ideal_below()
Fractional ideal (6*a + 2)

K.<a> = NumberField(2*x^2 - 1/3)
L.<b> = K.extension(5*x^2 + 1)
P = L.primes_above(2)[0]
P.ideal_below()

integral_basis()#

Return a basis for self as a $Z$ -module.

EXAMPLES:

sage: x = polygen(ZZ, 'x')
sage: K.<a,b> = NumberField([x^2 + 1, x^2 - 3])
sage: I = K.ideal(17*b - 3*a)
sage: x = I.integral_basis(); x  # random
[438, -b*a + 309, 219*a - 219*b, 156*a - 154*b]

x = polygen(ZZ, 'x')
K.<a,b> = NumberField([x^2 + 1, x^2 - 3])
I = K.ideal(17*b - 3*a)
x = I.integral_basis(); x  # random

The exact results are somewhat unpredictable, hence the # random flag, but we can test that they are indeed a basis:

sage: V, _, phi = K.absolute_vector_space()
sage: V.span([phi(u) for u in x], ZZ) == I.free_module()
True

V, _, phi = K.absolute_vector_space()
V.span([phi(u) for u in x], ZZ) == I.free_module()

integral_split()#

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: x = polygen(ZZ, 'x')
sage: K.<a, b> = NumberFieldTower([x^2 - 23, x^2 + 1])
sage: I = K.ideal([a + b/3])
sage: J, d = I.integral_split()
sage: J.is_integral()
True
sage: J == d*I
True

x = polygen(ZZ, 'x')
K.<a, b> = NumberFieldTower([x^2 - 23, x^2 + 1])
I = K.ideal([a + b/3])
J, d = I.integral_split()
J.is_integral()
J == d*I

is_integral()#

Return True if this ideal is integral.

EXAMPLES:

sage: x = polygen(ZZ, 'x')
sage: K.<a, b> = QQ.extension([x^2 + 11, x^2 - 5])
sage: I = K.ideal(7).prime_factors()[0]
sage: I.is_integral()
True
sage: (I/2).is_integral()
False

x = polygen(ZZ, 'x')
K.<a, b> = QQ.extension([x^2 + 11, x^2 - 5])
I = K.ideal(7).prime_factors()[0]
I.is_integral()
(I/2).is_integral()

is_prime()#

Return True if this ideal of a relative number field is prime.

EXAMPLES:

sage: x = polygen(ZZ, 'x')
sage: K.<a, b> = NumberField([x^2 - 17, x^3 - 2])
sage: K.ideal(a + b).is_prime()
True
sage: K.ideal(13).is_prime()
False

x = polygen(ZZ, 'x')
K.<a, b> = NumberField([x^2 - 17, x^3 - 2])
K.ideal(a + b).is_prime()
K.ideal(13).is_prime()

is_principal(proof=None)#

Return True if this ideal is principal. If so, set self.__reduced_generators, with length one.

EXAMPLES:

sage: x = polygen(ZZ, 'x')
sage: K.<a, b> = NumberField([x^2 - 23, x^2 + 1])
sage: I = K.ideal([7, (-1/2*b - 3/2)*a + 3/2*b + 9/2])
sage: I.is_principal()
True
sage: I # random
Fractional ideal ((1/2*b + 1/2)*a - 3/2*b - 3/2)

x = polygen(ZZ, 'x')
K.<a, b> = NumberField([x^2 - 23, x^2 + 1])
I = K.ideal([7, (-1/2*b - 3/2)*a + 3/2*b + 9/2])
I.is_principal()
I # random

is_zero()#

Return True if this is the zero ideal.

EXAMPLES:

sage: x = polygen(ZZ, 'x')
sage: K.<a, b> = NumberField([x^2 + 3, x^3 + 4])
sage: K.ideal(17).is_zero()
False
sage: K.ideal(0).is_zero()
True

x = polygen(ZZ, 'x')
K.<a, b> = NumberField([x^2 + 3, x^3 + 4])
K.ideal(17).is_zero()
K.ideal(0).is_zero()

norm()#

The norm of a fractional ideal in a relative number field is deliberately unimplemented, so that a user cannot mistake the absolute norm for the relative norm, or vice versa.

EXAMPLES:

sage: x = polygen(ZZ, 'x')
sage: K.<a, b> = NumberField([x^2 + 1, x^2 - 2])
sage: K.ideal(2).norm()
Traceback (most recent call last):
...
NotImplementedError: For a fractional ideal in a relative number field
you must use relative_norm or absolute_norm as appropriate

x = polygen(ZZ, 'x')
K.<a, b> = NumberField([x^2 + 1, x^2 - 2])
K.ideal(2).norm()

pari_rhnf()#

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

EXAMPLES:

sage: x = polygen(ZZ, 'x')
sage: K.<a, b> = NumberField([x^2 + 23, x^2 - 7])
sage: I = K.ideal(2, (a + 2*b + 3)/2)
sage: I.pari_rhnf()
[[1, -2; 0, 1], [[2, 1; 0, 1], 1/2]]

x = polygen(ZZ, 'x')
K.<a, b> = NumberField([x^2 + 23, x^2 - 7])
I = K.ideal(2, (a + 2*b + 3)/2)
I.pari_rhnf()

ramification_index()#

For ideals in relative number fields, ramification_index() is deliberately not implemented in order to avoid ambiguity. Either relative_ramification_index() or absolute_ramification_index() should be used instead.

EXAMPLES:

sage: x = polygen(ZZ, 'x')
sage: K.<a, b> = NumberField([x^2 + 1, x^2 - 2])
sage: K.ideal(2).ramification_index()
Traceback (most recent call last):
...
NotImplementedError: For an ideal in a relative number field you must use
relative_ramification_index or absolute_ramification_index as appropriate

x = polygen(ZZ, 'x')
K.<a, b> = NumberField([x^2 + 1, x^2 - 2])
K.ideal(2).ramification_index()

relative_norm()#

Compute the relative norm of this fractional ideal in a relative number field, returning an ideal in the base field.

EXAMPLES:

sage: R.<x> = QQ[]
sage: K.<a> = NumberField(x^2 + 6)
sage: L.<b> = K.extension(K['x'].gen()^4 + a)
sage: N = L.ideal(b).relative_norm(); N
Fractional ideal (-a)
sage: N.parent()
Monoid of ideals of Number Field in a with defining polynomial x^2 + 6
sage: N.ring()
Number Field in a with defining polynomial x^2 + 6
sage: PQ.<X> = QQ[]
sage: F.<a, b> = NumberField([X^2 - 2, X^2 - 3])
sage: PF.<Y> = F[]
sage: K.<c> = F.extension(Y^2 - (1 + a)*(a + b)*a*b)
sage: K.ideal(1).relative_norm()
Fractional ideal (1)
sage: K.ideal(13).relative_norm().relative_norm()
Fractional ideal (28561)
sage: K.ideal(13).relative_norm().relative_norm().relative_norm()
815730721
sage: K.ideal(13).absolute_norm()
815730721

R.<x> = QQ[]
K.<a> = NumberField(x^2 + 6)
L.<b> = K.extension(K['x'].gen()^4 + a)
N = L.ideal(b).relative_norm(); N
N.parent()
N.ring()
PQ.<X> = QQ[]
F.<a, b> = NumberField([X^2 - 2, X^2 - 3])
PF.<Y> = F[]
K.<c> = F.extension(Y^2 - (1 + a)*(a + b)*a*b)
K.ideal(1).relative_norm()
K.ideal(13).relative_norm().relative_norm()
K.ideal(13).relative_norm().relative_norm().relative_norm()
K.ideal(13).absolute_norm()

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

sage: K.<a> = NumberField(2*x^2 - 1/3)
sage: L.<b> = K.extension(5*x^2 + 1)
sage: P = L.primes_above(2)[0]
sage: P.relative_norm()
Fractional ideal (6*a + 2)

K.<a> = NumberField(2*x^2 - 1/3)
L.<b> = K.extension(5*x^2 + 1)
P = L.primes_above(2)[0]
P.relative_norm()

relative_ramification_index()#

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

The relative ramification index is the power of this prime appearing in the factorization of the prime ideal of the base field that this prime lies over.

Use absolute_ramification_index() to obtain the power of this prime occurring in the factorization of the rational prime that this prime lies over.

EXAMPLES:

sage: PQ.<X> = QQ[]
sage: F.<a, b> = NumberFieldTower([X^2 - 2, X^2 - 3])
sage: PF.<Y> = F[]
sage: K.<c> = F.extension(Y^2 - (1 + a)*(a + b)*a*b)
sage: I = K.ideal(3, c)
sage: I.relative_ramification_index()
2
sage: I.ideal_below()  # random sign
Fractional ideal (b)
sage: I.ideal_below() == K.ideal(b)
True
sage: K.ideal(b) == I^2
True

PQ.<X> = QQ[]
F.<a, b> = NumberFieldTower([X^2 - 2, X^2 - 3])
PF.<Y> = F[]
K.<c> = F.extension(Y^2 - (1 + a)*(a + b)*a*b)
I = K.ideal(3, c)
I.relative_ramification_index()
I.ideal_below()  # random sign
I.ideal_below() == K.ideal(b)
K.ideal(b) == I^2

residue_class_degree()#

Return the residue class degree of this prime.

EXAMPLES:

sage: PQ.<X> = QQ[]
sage: F.<a, b> = NumberFieldTower([X^2 - 2, X^2 - 3])
sage: PF.<Y> = F[]
sage: K.<c> = F.extension(Y^2 - (1 + a)*(a + b)*a*b)
sage: [I.residue_class_degree() for I in K.ideal(c).prime_factors()]
[1, 2]

PQ.<X> = QQ[]
F.<a, b> = NumberFieldTower([X^2 - 2, X^2 - 3])
PF.<Y> = F[]
K.<c> = F.extension(Y^2 - (1 + a)*(a + b)*a*b)
[I.residue_class_degree() for I in K.ideal(c).prime_factors()]

residues()#

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

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

EXAMPLES:

sage: x = polygen(ZZ, 'x')
sage: K.<a, w> = NumberFieldTower([x^2 - 3, x^2 + x + 1])
sage: I = K.ideal(6, -w*a - w + 4)
sage: list(I.residues())[:5]
[(25/3*w - 1/3)*a + 22*w + 1,
(16/3*w - 1/3)*a + 13*w,
(7/3*w - 1/3)*a + 4*w - 1,
(-2/3*w - 1/3)*a - 5*w - 2,
(-11/3*w - 1/3)*a - 14*w - 3]

x = polygen(ZZ, 'x')
K.<a, w> = NumberFieldTower([x^2 - 3, x^2 + x + 1])
I = K.ideal(6, -w*a - w + 4)
list(I.residues())[:5]

smallest_integer()#

Return the smallest non-negative integer in $I \cap Z$ , where $I$ is this ideal. If $I = 0$ , returns $0$ .

EXAMPLES:

sage: x = polygen(ZZ, 'x')
sage: K.<a, b> = NumberFieldTower([x^2 - 23, x^2 + 1])
sage: I = K.ideal([a + b])
sage: I.smallest_integer()
12
sage: [m for m in range(13) if m in I]
[0, 12]

x = polygen(ZZ, 'x')
K.<a, b> = NumberFieldTower([x^2 - 23, x^2 + 1])
I = K.ideal([a + b])
I.smallest_integer()
[m for m in range(13) if m in I]

valuation(p)#

Return the valuation of this fractional ideal at $p$ .

INPUT:

p – a prime ideal $p$ of this relative number field.

OUTPUT:

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

EXAMPLES:

sage: x = polygen(ZZ, 'x')
sage: K.<a, b> = NumberField([x^2 - 17, x^3 - 2])
sage: A = K.ideal(a + b)
sage: A.is_prime()
True
sage: (A*K.ideal(3)).valuation(A)
1
sage: K.ideal(25).valuation(5)
Traceback (most recent call last):
...
ValueError: p (= Fractional ideal (5)) must be a prime

x = polygen(ZZ, 'x')
K.<a, b> = NumberField([x^2 - 17, x^3 - 2])
A = K.ideal(a + b)
A.is_prime()
(A*K.ideal(3)).valuation(A)
K.ideal(25).valuation(5)

sage.rings.number_field.number_field_ideal_rel.is_NumberFieldFractionalIdeal_rel(x)#

Return True if $x$ is a fractional ideal of a relative number field.

EXAMPLES:

sage: from sage.rings.number_field.number_field_ideal_rel import is_NumberFieldFractionalIdeal_rel
sage: from sage.rings.number_field.number_field_ideal import is_NumberFieldFractionalIdeal
sage: is_NumberFieldFractionalIdeal_rel(2/3)
False
sage: is_NumberFieldFractionalIdeal_rel(ideal(5))
False
sage: x = polygen(ZZ, 'x')
sage: k.<a> = NumberField(x^2 + 2)
sage: I = k.ideal([a + 1]); I
Fractional ideal (a + 1)
sage: is_NumberFieldFractionalIdeal_rel(I)
False
sage: R.<x> = QQ[]
sage: K.<a> = NumberField(x^2 + 6)
sage: L.<b> = K.extension(K['x'].gen()^4 + a)
sage: I = L.ideal(b); I
Fractional ideal (6, b)
sage: is_NumberFieldFractionalIdeal_rel(I)
True
sage: N = I.relative_norm(); N
Fractional ideal (-a)
sage: is_NumberFieldFractionalIdeal_rel(N)
False
sage: is_NumberFieldFractionalIdeal(N)
True

from sage.rings.number_field.number_field_ideal_rel import is_NumberFieldFractionalIdeal_rel
from sage.rings.number_field.number_field_ideal import is_NumberFieldFractionalIdeal
is_NumberFieldFractionalIdeal_rel(2/3)
is_NumberFieldFractionalIdeal_rel(ideal(5))
x = polygen(ZZ, 'x')
k.<a> = NumberField(x^2 + 2)
I = k.ideal([a + 1]); I
is_NumberFieldFractionalIdeal_rel(I)
R.<x> = QQ[]
K.<a> = NumberField(x^2 + 6)
L.<b> = K.extension(K['x'].gen()^4 + a)
I = L.ideal(b); I
is_NumberFieldFractionalIdeal_rel(I)
N = I.relative_norm(); N
is_NumberFieldFractionalIdeal_rel(N)
is_NumberFieldFractionalIdeal(N)