Ideals of relative number fields

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

AUTHORS:

• Steven Sivek (2005-05-16)

• William Stein (2007-09-06)

• Nick Alexander (2009-01)

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

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 \(L/\QQ\).

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()

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)

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

Now pass \(m\) as the name for the generator of the absolute field:

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

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

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

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

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

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]

free_module()

Return this ideal as a \(\ZZ\)-submodule of the \(\QQ\)-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

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,)

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

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

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

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

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

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)

integral_basis()

Return a basis for self as a \(\ZZ\)-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]

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

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

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

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

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)

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

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

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]]

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

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

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)

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

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]

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]

smallest_integer()

Return the smallest non-negative integer in \(I \cap \ZZ\), 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]

valuation(p)

Return the valuation of this fractional ideal at \(\mathfrak{p}\).

INPUT:

• p – a prime ideal \(\mathfrak{p}\) of this relative 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: 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

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