Class NumberFieldFractionalIdeal_rel

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

EXAMPLES:

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

Note that if the ideal is not principal, then this uses PARI's
\code{idealtwoelt} function, which takes exponential time, the
first time it is called for each ideal.  Also, this indirectly
uses \code{bnfisprincipal}, so set \code{proof=True} if you
want to prove correctness (which \emph{is} the default).

EXAMPLE:
    sage: R.<x> = PolynomialRing(QQ)
    sage: K.<i> = NumberField(x^2+1, 'i')
    sage: J = K.ideal([i+1, 2])
    sage: J.gens()
    (i + 1, 2)
    sage: J.gens_reduced()
    (i + 1,)

TESTS:
    sage: all(j.parent() is K for j in J.gens())
    True
    sage: all(j.parent() is K for j in J.gens_reduced())
    True

    sage: K.<a> = NumberField(x^4 + 10*x^2 + 20)
    sage: J = K.prime_above(5)
    sage: J.is_principal()
    False
    sage: J.gens_reduced()
    (5, a)
    sage: all(j.parent() is K for j in J.gens())
    True
    sage: all(j.parent() is K for j in J.gens_reduced())
    True

Overrides: number_field_ideal.NumberFieldIdeal.gens_reduced

(inherited documentation)

Return the multiplicative inverse of self.  Call with ~self.

EXAMPLES:
    sage: K.<a,b> = NumberField([x^2 + 1, x^2 + 2])
    sage: I = K.fractional_ideal(4)
    sage: I^(-1)
    Fractional ideal (1/4)
    sage: I * I^(-1)
    Fractional ideal (1)

Overrides: number_field_ideal.NumberFieldFractionalIdeal.__invert__

Return True if this ideal is principal.

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

EXAMPLES:
We create equal ideals in two different ways, and note that
they are both actually principal ideals. 
    sage: K = QuadraticField(-119,'a')
    sage: P = K.ideal([2]).factor()[1][0]
    sage: I = P^5
    sage: I.is_principal()
    True

Overrides: number_field_ideal.NumberFieldIdeal.is_principal

(inherited documentation)

Return True iff self is the zero ideal

EXAMPLES:
    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

    (0 is a NumberFieldIdeal, not a NumberFieldFractionIdeal)

Overrides: number_field_ideal.NumberFieldIdeal.is_zero

(inherited documentation)

Compute the relative norm of this extension L/K as an ideal of K.

EXAMPLE:
    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).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

Overrides: number_field_ideal.NumberFieldIdeal.norm

Factorization of this ideal in terms of prime ideals. 

EXAMPLES:
    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 (a^2 + 2*a + 2)) * (Fractional ideal (a^2 - 2*a + 2))
    sage: type(F)
    <class 'sage.structure.factorization.Factorization'>
    sage: list(F)
    [(Fractional ideal (a^2 + 2*a + 2), 1), (Fractional ideal (a^2 - 2*a + 2), 1)]
    sage: F.prod()
    Fractional ideal (19)

Overrides: number_field_ideal.NumberFieldFractionalIdeal.factor

(inherited documentation)

Return a list of generators for this ideal as a $\mathbb{Z}$-module.

EXAMPLE:
    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]

Overrides: number_field_ideal.NumberFieldIdeal.integral_basis

(inherited documentation)

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.

EXAMPLE:
    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

Overrides: number_field_ideal.NumberFieldIdeal.integral_split

(inherited documentation)

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

EXAMPLES:
    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        

Overrides: number_field_ideal.NumberFieldFractionalIdeal.is_maximal

(inherited documentation)

Return True if this ideal is prime.

EXAMPLES:
    sage: K.<a> = NumberField(x^2 - 17); K
    Number Field in a with defining polynomial x^2 - 17
    sage: K.ideal(5).is_prime()
    True
    sage: K.ideal(13).is_prime()
    False
    sage: K.ideal(17).is_prime()
    False        

Overrides: number_field_ideal.NumberFieldIdeal.is_prime

(inherited documentation)

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: 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]        

Overrides: number_field_ideal.NumberFieldFractionalIdeal.residue_class_degree

(inherited documentation)

Return the smallest nonnegative integer in $I \cap \mathbb{Z}$,
where $I$ is this ideal.  If $I = 0$, raise a ValueError.

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

Overrides: number_field_ideal.NumberFieldIdeal.smallest_integer

(inherited documentation)

Return the valuation of this fractional ideal at the prime
$\mathfrak{p}$.  If $\mathfrak{p}$ is not prime, raise a
ValueError.

INPUT:
    p -- a prime ideal of this number field.

OUTPUT:
    integer

EXAMPLES:
    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 (= 0) must be nonzero

Overrides: number_field_ideal.NumberFieldIdeal.valuation

(inherited documentation)