Ideals of function fields

Ideals of an order of a function field include all fractional ideals of the order. Sage provides basic arithmetic with fractional ideals.

The fractional ideals of the maximal order of a global function field forms a multiplicative monoid. Sage allows advanced arithmetic with the fractional ideals. For example, an ideal of the maximal order can be factored into a product of prime ideals.

EXAMPLES:

Ideals in the maximal order of a rational function field:

sage: K.<x> = FunctionField(QQ)
sage: O = K.maximal_order()
sage: I = O.ideal(x^3 + 1); I
Ideal (x^3 + 1) of Maximal order of Rational function field in x over Rational Field
sage: I^2
Ideal (x^6 + 2*x^3 + 1) of Maximal order of Rational function field in x over Rational Field
sage: ~I
Ideal (1/(x^3 + 1)) of Maximal order of Rational function field in x over Rational Field
sage: ~I * I
Ideal (1) of Maximal order of Rational function field in x over Rational Field

Ideals in the equation order of an extension of a rational function field:

sage: K.<x> = FunctionField(QQ); R.<y> = K[]
sage: L.<y> = K.extension(y^2 - x^3 - 1)
sage: O = L.equation_order()
sage: I = O.ideal(y); I
Ideal (x^3 + 1, -y) of Order in Function field in y defined by y^2 - x^3 - 1
sage: I^2
Ideal (x^3 + 1, (-x^3 - 1)*y) of Order in Function field in y defined by y^2 - x^3 - 1

Ideals in the maximal order of a global function field:

sage: K.<x> = FunctionField(GF(2)); R.<y> = K[]
sage: L.<y> = K.extension(y^2 - x^3*y - x)
sage: O = L.maximal_order()
sage: I = O.ideal(y)
sage: I^2
Ideal (x) of Maximal order of Function field in y defined by y^2 + x^3*y + x
sage: ~I
Ideal (1/x*y) of Maximal order of Function field in y defined by y^2 + x^3*y + x
sage: ~I * I
Ideal (1) of Maximal order of Function field in y defined by y^2 + x^3*y + x

sage: J = O.ideal(x+y) * I
sage: J.factor()
(Ideal (y) of Maximal order of Function field in y defined by y^2 + x^3*y + x)^2 *
(Ideal (x^3 + x + 1, y + x) of Maximal order of Function field in y defined by y^2 + x^3*y + x)

Ideals in the maximal infinite order of a global function field:

sage: K.<x> = FunctionField(GF(3^2)); R.<t> = K[]
sage: F.<y> = K.extension(t^3 + t^2 - x^4)
sage: Oinf = F.maximal_order_infinite()
sage: I = Oinf.ideal(1/y)
sage: I + I == I
True
sage: I^2
Ideal (1/x^4*y) of Maximal infinite order of Function field in y defined by y^3 + y^2 + 2*x^4
sage: ~I
Ideal (y) of Maximal infinite order of Function field in y defined by y^3 + y^2 + 2*x^4
sage: ~I * I
Ideal (1) of Maximal infinite order of Function field in y defined by y^3 + y^2 + 2*x^4
sage: I.factor()
(Ideal (1/x^3*y^2) of Maximal infinite order of Function field in y defined by y^3 + y^2 + 2*x^4)^4

AUTHORS:

• William Stein (2010): initial version

• Maarten Derickx (2011-09-14): fixed ideal_with_gens_over_base()

• Kwankyu Lee (2017-04-30): added ideals for global function fields

class sage.rings.function_field.ideal.FunctionFieldIdeal(ring)

Bases: sage.structure.element.Element

Base class of fractional ideals of function fields.

INPUT:

• ring – ring of the ideal

EXAMPLES:

sage: K.<x> = FunctionField(GF(7))
sage: O = K.equation_order()
sage: O.ideal(x^3+1)
Ideal (x^3 + 1) of Maximal order of Rational function field in x over Finite Field of size 7

base_ring()

Return the base ring of this ideal.

EXAMPLES:

sage: K.<x> = FunctionField(QQ); R.<y> = K[]
sage: L.<y> = K.extension(y^2 - x^3 - 1)
sage: O = L.equation_order()
sage: I = O.ideal(x^2 + 1)
sage: I.base_ring()
Order in Function field in y defined by y^2 - x^3 - 1

divisor()

Return the divisor corresponding to the ideal.

EXAMPLES:

sage: K.<x> = FunctionField(GF(4))
sage: O = K.maximal_order()
sage: I = O.ideal(x*(x + 1)^2/(x^2 + x + 1))
sage: I.divisor()
Place (x) + 2*Place (x + 1) - Place (x + z2) - Place (x + z2 + 1)

sage: Oinf = K.maximal_order_infinite()
sage: I = Oinf.ideal((x + 1)/(x^3 + 1))
sage: I.divisor()
2*Place (1/x)

sage: K.<x> = FunctionField(GF(2)); _.<T> = PolynomialRing(K)
sage: F.<y> = K.extension(T^3 - x^2*(x^2 + x + 1)^2)
sage: O = F.maximal_order()
sage: I = O.ideal(y)
sage: I.divisor()
2*Place (x, (1/(x^3 + x^2 + x))*y^2)
 + 2*Place (x^2 + x + 1, (1/(x^3 + x^2 + x))*y^2)

sage: Oinf = F.maximal_order_infinite()
sage: I = Oinf.ideal(y)
sage: I.divisor()
-2*Place (1/x, 1/x^4*y^2 + 1/x^2*y + 1)
 - 2*Place (1/x, 1/x^2*y + 1)

sage: K.<x> = FunctionField(GF(2)); _.<Y> = K[]
sage: L.<y> = K.extension(Y^2 + Y + x + 1/x)
sage: O = L.maximal_order()
sage: I = O.ideal(y)
sage: I.divisor()
- Place (x, x*y)
 + 2*Place (x + 1, x*y)

sage: Oinf = L.maximal_order_infinite()
sage: I = Oinf.ideal(y)
sage: I.divisor()
- Place (1/x, 1/x*y)

divisor_of_poles()

Return the divisor of poles corresponding to the ideal.

EXAMPLES:

sage: K.<x> = FunctionField(GF(4))
sage: O = K.maximal_order()
sage: I = O.ideal(x*(x + 1)^2/(x^2 + x + 1))
sage: I.divisor_of_poles()
Place (x + z2) + Place (x + z2 + 1)

sage: K.<x> = FunctionField(GF(2))
sage: Oinf = K.maximal_order_infinite()
sage: I = Oinf.ideal((x + 1)/(x^3 + 1))
sage: I.divisor_of_poles()
0

sage: K.<x> = FunctionField(GF(2)); _.<Y> = K[]
sage: L.<y> = K.extension(Y^2 + Y + x + 1/x)
sage: O = L.maximal_order()
sage: I = O.ideal(y)
sage: I.divisor_of_poles()
Place (x, x*y)

divisor_of_zeros()

Return the divisor of zeros corresponding to the ideal.

EXAMPLES:

sage: K.<x> = FunctionField(GF(4))
sage: O = K.maximal_order()
sage: I = O.ideal(x*(x + 1)^2/(x^2 + x + 1))
sage: I.divisor_of_zeros()
Place (x) + 2*Place (x + 1)

sage: K.<x> = FunctionField(GF(2))
sage: Oinf = K.maximal_order_infinite()
sage: I = Oinf.ideal((x + 1)/(x^3 + 1))
sage: I.divisor_of_zeros()
2*Place (1/x)

sage: K.<x> = FunctionField(GF(2)); _.<Y> = K[]
sage: L.<y> = K.extension(Y^2 + Y + x + 1/x)
sage: O = L.maximal_order()
sage: I = O.ideal(y)
sage: I.divisor_of_zeros()
2*Place (x + 1, x*y)

factor()

Return the factorization of this ideal.

Subclass of this class should define _factor() method that returns a list of prime ideal and multiplicity pairs.

EXAMPLES:

sage: K.<x> = FunctionField(GF(4))
sage: O = K.maximal_order()
sage: I = O.ideal(x^3*(x + 1)^2)
sage: I.factor()
(Ideal (x) of Maximal order of Rational function field in x
over Finite Field in z2 of size 2^2)^3 *
(Ideal (x + 1) of Maximal order of Rational function field in x
over Finite Field in z2 of size 2^2)^2

sage: Oinf = K.maximal_order_infinite()
sage: I = Oinf.ideal((x + 1)/(x^3 + 1))
sage: I.factor()
(Ideal (1/x) of Maximal infinite order of Rational function field in x
over Finite Field in z2 of size 2^2)^2

sage: K.<x> = FunctionField(GF(2)); _.<T> = PolynomialRing(K)
sage: F.<y> = K.extension(T^3 - x^2*(x^2 + x + 1)^2)
sage: O = F.maximal_order()
sage: I = O.ideal(y)
sage: I == I.factor().prod()
True

sage: Oinf = F.maximal_order_infinite()
sage: f= 1/x
sage: I = Oinf.ideal(f)
sage: I.factor()
(Ideal ((1/(x^4 + x^3 + x^2))*y^2 + 1/x^2*y + 1) of Maximal infinite order
of Function field in y defined by y^3 + x^6 + x^4 + x^2) *
(Ideal ((1/(x^4 + x^3 + x^2))*y^2 + 1) of Maximal infinite order
of Function field in y defined by y^3 + x^6 + x^4 + x^2)

sage: K.<x> = FunctionField(QQ); _.<Y> = K[]
sage: F.<y> = K.extension(Y^3 - x^2*(x^2 + x + 1)^2)
sage: O = F.maximal_order()
sage: I = O.ideal(y)
sage: I == I.factor().prod()
True

sage: K.<x> = FunctionField(QQ); _.<Y> = K[]
sage: L.<y> = K.extension(Y^2 + Y + x + 1/x)
sage: O = L.maximal_order()
sage: I = O.ideal(y)
sage: I == I.factor().prod()
True

gens_reduced()

Return reduced generators.

For now, this method just looks at the generators and sees if any can be removed without changing the ideal. It prefers principal representations (a single generator) over all others, and otherwise picks the generator set with the shortest print representation.

This method is provided so that ideals in function fields have the method gens_reduced(), just like ideals of number fields. Sage linear algebra machinery sometimes requires this.

EXAMPLES:

sage: K.<x> = FunctionField(GF(7))
sage: O = K.equation_order()
sage: I = O.ideal(x,x^2,x^2+x)
sage: I.gens_reduced()
(x,)

place()

Return the place associated with this prime ideal.

EXAMPLES:

sage: K.<x> = FunctionField(GF(4))
sage: O = K.maximal_order()
sage: I = O.ideal(x^2 + x + 1)
sage: I.place()
Traceback (most recent call last):
...
TypeError: not a prime ideal
sage: I = O.ideal(x^3+x+1)
sage: I.place()
Place (x^3 + x + 1)

sage: K.<x> = FunctionField(GF(2))
sage: Oinf = K.maximal_order_infinite()
sage: I = Oinf.ideal((x + 1)/(x^3 + 1))
sage: p = I.factor()[0][0]
sage: p.place()
Place (1/x)

sage: K.<x> = FunctionField(GF(2)); _.<t> = PolynomialRing(K)
sage: F.<y> = K.extension(t^3-x^2*(x^2+x+1)^2)
sage: O = F.maximal_order()
sage: I = O.ideal(y)
sage: [f.place() for f,_ in I.factor()]
[Place (x, (1/(x^3 + x^2 + x))*y^2),
 Place (x^2 + x + 1, (1/(x^3 + x^2 + x))*y^2)]

sage: K.<x> = FunctionField(GF(2)); _.<Y> = K[]
sage: L.<y> = K.extension(Y^2 + Y + x + 1/x)
sage: O = L.maximal_order()
sage: I = O.ideal(y)
sage: [f.place() for f,_ in I.factor()]
[Place (x, x*y), Place (x + 1, x*y)]

sage: K.<x> = FunctionField(GF(3^2)); R.<t> = PolynomialRing(K)
sage: F.<y> = K.extension(t^3 + t^2 - x^4)
sage: Oinf = F.maximal_order_infinite()
sage: I = Oinf.ideal(1/x)
sage: I.factor()
(Ideal (1/x^3*y^2) of Maximal infinite order of Function field
in y defined by y^3 + y^2 + 2*x^4)^3
sage: J = I.factor()[0][0]
sage: J.is_prime()
True
sage: J.place()
Place (1/x, 1/x^3*y^2)

sage: K.<x> = FunctionField(GF(2)); _.<Y> = K[]
sage: L.<y> = K.extension(Y^2 + Y + x + 1/x)
sage: Oinf = L.maximal_order_infinite()
sage: I = Oinf.ideal(1/x)
sage: I.factor()
(Ideal (1/x*y) of Maximal infinite order of Function field in y
defined by y^2 + y + (x^2 + 1)/x)^2
sage: J = I.factor()[0][0]
sage: J.is_prime()
True
sage: J.place()
Place (1/x, 1/x*y)

ring()

Return the ring to which this ideal belongs.

EXAMPLES:

sage: K.<x> = FunctionField(GF(7))
sage: O = K.equation_order()
sage: I = O.ideal(x,x^2,x^2+x)
sage: I.ring()
Maximal order of Rational function field in x over Finite Field of size 7

class sage.rings.function_field.ideal.FunctionFieldIdealInfinite(ring)

Bases: sage.rings.function_field.ideal.FunctionFieldIdeal

Base class of ideals of maximal infinite orders

class sage.rings.function_field.ideal.FunctionFieldIdealInfinite_module(ring, module)

Bases: sage.rings.function_field.ideal.FunctionFieldIdealInfinite, sage.rings.ideal.Ideal_generic

A fractional ideal specified by a finitely generated module over the integers of the base field.

INPUT:

• ring – order in a function field

• module – module

EXAMPLES:

sage: K.<x> = FunctionField(QQ); R.<y> = K[]
sage: L.<y> = K.extension(y^2 - x^3 - 1)
sage: O = L.equation_order()
sage: O.ideal(y)
Ideal (x^3 + 1, -y) of Order in Function field in y defined by y^2 - x^3 - 1

module()

Return the module over the maximal order of the base field that underlies this ideal.

The formation of the module is compatible with the vector space corresponding to the function field.

EXAMPLES:

sage: K.<x> = FunctionField(GF(7))
sage: O = K.maximal_order(); O
Maximal order of Rational function field in x over Finite Field of size 7
sage: K.polynomial_ring()
Univariate Polynomial Ring in x over Rational function field in x over Finite Field of size 7
sage: I = O.ideal([x^2 + 1, x*(x^2+1)])
sage: I.gens()
(x^2 + 1,)
sage: I.module()
Free module of degree 1 and rank 1 over Maximal order of Rational function field in x over Finite Field of size 7
Echelon basis matrix:
[x^2 + 1]
sage: V, from_V, to_V = K.vector_space(); V
Vector space of dimension 1 over Rational function field in x over Finite Field of size 7
sage: I.module().is_submodule(V)
True

class sage.rings.function_field.ideal.FunctionFieldIdealInfinite_polymod(ring, ideal)

Bases: sage.rings.function_field.ideal.FunctionFieldIdealInfinite

Ideals of the infinite maximal order of an algebraic function field.

INPUT:

• ring – infinite maximal order of the function field

• ideal – ideal in the inverted function field

EXAMPLES:

sage: K.<x> = FunctionField(GF(3^2)); R.<t> = PolynomialRing(K)
sage: F.<y> = K.extension(t^3+t^2-x^4)
sage: Oinf = F.maximal_order_infinite()
sage: Oinf.ideal(1/y)
Ideal (1/x^4*y^2) of Maximal infinite order of Function field
in y defined by y^3 + y^2 + 2*x^4

gens()

Return a set of generators of this ideal.

EXAMPLES:

sage: K.<x> = FunctionField(GF(3^2)); R.<t> = PolynomialRing(K)
sage: F.<y> = K.extension(t^3+t^2-x^4)
sage: Oinf = F.maximal_order_infinite()
sage: I = Oinf.ideal(x+y)
sage: I.gens()
(x, y, 1/x^2*y^2)

sage: K.<x> = FunctionField(GF(2)); _.<Y> = K[]
sage: L.<y> = K.extension(Y^2 + Y + x + 1/x)
sage: Oinf = L.maximal_order_infinite()
sage: I = Oinf.ideal(x+y)
sage: I.gens()
(x, y)

gens_over_base()

Return a set of generators of this ideal.

EXAMPLES:

sage: K.<x> = FunctionField(GF(3^2)); _.<t> = K[]
sage: F.<y> = K.extension(t^3 + t^2 - x^4)
sage: Oinf = F.maximal_order_infinite()
sage: I = Oinf.ideal(x + y)
sage: I.gens_over_base()
(x, y, 1/x^2*y^2)

gens_two()

Return a set of at most two generators of this ideal.

EXAMPLES:

sage: K.<x> = FunctionField(GF(3^2)); R.<t> = PolynomialRing(K)
sage: F.<y> = K.extension(t^3+t^2-x^4)
sage: Oinf = F.maximal_order_infinite()
sage: I = Oinf.ideal(x+y)
sage: I.gens_two()
(x, y)

sage: K.<x> = FunctionField(GF(2)); _.<Y> = K[]
sage: L.<y> = K.extension(Y^2+Y+x+1/x)
sage: Oinf = L.maximal_order_infinite()
sage: I = Oinf.ideal(x+y)
sage: I.gens_two()
(x,)

ideal_below()

Return a set of generators of this ideal.

EXAMPLES:

sage: K.<x> = FunctionField(GF(3^2)); _.<t> = K[]
sage: F.<y> = K.extension(t^3 + t^2 - x^4)
sage: Oinf = F.maximal_order_infinite()
sage: I = Oinf.ideal(1/y^2)
sage: I.ideal_below()
Ideal (x^3) of Maximal order of Rational function field
in x over Finite Field in z2 of size 3^2

is_prime()

Return True if this ideal is a prime ideal.

EXAMPLES:

sage: K.<x> = FunctionField(GF(3^2)); _.<t> = PolynomialRing(K)
sage: F.<y> = K.extension(t^3 + t^2 - x^4)
sage: Oinf = F.maximal_order_infinite()
sage: I = Oinf.ideal(1/x)
sage: I.factor()
(Ideal (1/x^3*y^2) of Maximal infinite order of Function field
in y defined by y^3 + y^2 + 2*x^4)^3
sage: I.is_prime()
False
sage: J = I.factor()[0][0]
sage: J.is_prime()
True

sage: K.<x> = FunctionField(GF(2)); _.<Y> = K[]
sage: L.<y> = K.extension(Y^2 + Y + x + 1/x)
sage: Oinf = L.maximal_order_infinite()
sage: I = Oinf.ideal(1/x)
sage: I.factor()
(Ideal (1/x*y) of Maximal infinite order of Function field in y
defined by y^2 + y + (x^2 + 1)/x)^2
sage: I.is_prime()
False
sage: J = I.factor()[0][0]
sage: J.is_prime()
True

prime_below()

Return the prime of the base order that underlies this prime ideal.

EXAMPLES:

sage: K.<x> = FunctionField(GF(3^2)); _.<t> = PolynomialRing(K)
sage: F.<y> = K.extension(t^3+t^2-x^4)
sage: Oinf = F.maximal_order_infinite()
sage: I = Oinf.ideal(1/x)
sage: I.factor()
(Ideal (1/x^3*y^2) of Maximal infinite order of Function field
in y defined by y^3 + y^2 + 2*x^4)^3
sage: J = I.factor()[0][0]
sage: J.is_prime()
True
sage: J.prime_below()
Ideal (1/x) of Maximal infinite order of Rational function field
in x over Finite Field in z2 of size 3^2

sage: K.<x> = FunctionField(GF(2)); _.<Y> = K[]
sage: L.<y> = K.extension(Y^2 + Y + x + 1/x)
sage: Oinf = L.maximal_order_infinite()
sage: I = Oinf.ideal(1/x)
sage: I.factor()
(Ideal (1/x*y) of Maximal infinite order of Function field in y
defined by y^2 + y + (x^2 + 1)/x)^2
sage: J = I.factor()[0][0]
sage: J.is_prime()
True
sage: J.prime_below()
Ideal (1/x) of Maximal infinite order of Rational function field in x
over Finite Field of size 2

valuation(ideal)

Return the valuation of ideal with respect to this prime ideal.

INPUT:

• ideal – fractional ideal

EXAMPLES:

sage: K.<x>=FunctionField(GF(2)); _.<Y> = K[]
sage: L.<y>=K.extension(Y^2 + Y + x + 1/x)
sage: Oinf = L.maximal_order_infinite()
sage: I = Oinf.ideal(y)
sage: [f.valuation(I) for f,_ in I.factor()]
[-1]

class sage.rings.function_field.ideal.FunctionFieldIdealInfinite_rational(ring, gen)

Bases: sage.rings.function_field.ideal.FunctionFieldIdealInfinite

Fractional ideal of the maximal order of rational function field.

INPUT:

• ring – infinite maximal order

• gen– generator

Note that the infinite maximal order is a principal ideal domain.

EXAMPLES:

sage: K.<x> = FunctionField(GF(2))
sage: Oinf = K.maximal_order_infinite()
sage: Oinf.ideal(x)
Ideal (x) of Maximal infinite order of Rational function field in x over Finite Field of size 2

gen()

Return the generator of this principal ideal.

EXAMPLES:

sage: K.<x> = FunctionField(GF(2))
sage: Oinf = K.maximal_order_infinite()
sage: I = Oinf.ideal((x+1)/(x^3+x),(x^2+1)/x^4)
sage: I.gen()
1/x^2

gens()

Return the generator of this principal ideal.

EXAMPLES:

sage: K.<x> = FunctionField(GF(2))
sage: Oinf = K.maximal_order_infinite()
sage: I = Oinf.ideal((x+1)/(x^3+x),(x^2+1)/x^4)
sage: I.gens()
(1/x^2,)

gens_over_base()

Return the generator of this ideal as a rank one module over the infinite maximal order.

EXAMPLES:

sage: K.<x> = FunctionField(GF(2))
sage: Oinf = K.maximal_order_infinite()
sage: I = Oinf.ideal((x+1)/(x^3+x),(x^2+1)/x^4)
sage: I.gens_over_base()
(1/x^2,)

is_prime()

Return True if this ideal is a prime ideal.

EXAMPLES:

sage: K.<x> = FunctionField(GF(2))
sage: Oinf = K.maximal_order_infinite()
sage: I = Oinf.ideal(x/(x^2 + 1))
sage: I.is_prime()
True

valuation(ideal)

Return the valuation of ideal at this prime ideal.

INPUT:

• ideal – fractional ideal

EXAMPLES:

sage: F.<x> = FunctionField(QQ)
sage: O = F.maximal_order_infinite()
sage: p = O.ideal(1/x)
sage: p.valuation(O.ideal(x/(x+1)))
0
sage: p.valuation(O.ideal(0))
+Infinity

class sage.rings.function_field.ideal.FunctionFieldIdeal_global(ring, hnf, denominator=1)

Bases: sage.rings.function_field.ideal.FunctionFieldIdeal_polymod

Fractional ideals of canonical function fields

INPUT:

• ring – order in a function field

• hnf – matrix in hermite normal form

• denominator – denominator

The rows of hnf is a basis of the ideal, which itself is denominator times the fractional ideal.

EXAMPLES:

sage: K.<x> = FunctionField(GF(2)); R.<y> = K[]
sage: L.<y> = K.extension(y^2 - x^3*y - x)
sage: O = L.maximal_order()
sage: O.ideal(y)
Ideal (y) of Maximal order of Function field in y defined by y^2 + x^3*y + x

gens()

Return a set of generators of this ideal.

This provides whatever set of generators as quickly as possible.

EXAMPLES:

sage: K.<x> = FunctionField(GF(2)); _.<Y> = K[]
sage: L.<y> = K.extension(Y^2 - x^3*Y - x)
sage: O = L.maximal_order()
sage: I = O.ideal(x+y)
sage: I.gens()
(x^4 + x^2 + x, y + x)

sage: L.<y> = K.extension(Y^2 +Y + x + 1/x)
sage: O = L.maximal_order()
sage: I = O.ideal(x+y)
sage: I.gens()
(x^3 + 1, y + x)

gens_two()

Return two generators of this fractional ideal.

If the ideal is principal, one generator may be returned.

ALGORITHM:

At most two generators are required to generate ideals in Dedekind domains.

Lemma 4.7.9, algorithm 4.7.10, and exercise 4.29 of [Coh1993] tell us that for an integral ideal \(I\) in a number field, if we pick \(a\) such that \(\gcd(N(I), N(a)/N(I)) = 1\), then \(a\) and \(N(I)\) generate the ideal. \(N()\) is the norm, and this result (presumably) generalizes to function fields.

After computing \(N(I)\), we search exhaustively to find \(a\).

Todo

Always return a single generator for a principal ideal.

Testing for principality is not trivial. Algorithm 6.5.10 of [Coh1993] could probably be adapted for function fields.

EXAMPLES:

sage: K.<x> = FunctionField(GF(2)); _.<t> = K[]
sage: F.<y> = K.extension(t^3 - x^2*(x^2 + x + 1)^2)
sage: O = F.maximal_order()
sage: I = O.ideal(y)
sage: I  # indirect doctest
Ideal (y) of Maximal order of Function field
in y defined by y^3 + x^6 + x^4 + x^2
sage: ~I  # indirect doctest
Ideal ((1/(x^6 + x^4 + x^2))*y^2) of Maximal order of Function field
in y defined by y^3 + x^6 + x^4 + x^2

sage: K.<x> = FunctionField(GF(2)); _.<Y> = K[]
sage: L.<y> = K.extension(Y^2 + Y + x + 1/x)
sage: O = L.maximal_order()
sage: I = O.ideal(y)
sage: I  # indirect doctest
Ideal (y) of Maximal order of Function field in y
defined by y^2 + y + (x^2 + 1)/x
sage: ~I  # indirect doctest
Ideal ((x/(x^2 + 1))*y + x/(x^2 + 1)) of Maximal order
of Function field in y defined by y^2 + y + (x^2 + 1)/x

class sage.rings.function_field.ideal.FunctionFieldIdeal_module(ring, module)

Bases: sage.rings.function_field.ideal.FunctionFieldIdeal, sage.rings.ideal.Ideal_generic

A fractional ideal specified by a finitely generated module over the integers of the base field.

INPUT:

• ring – an order in a function field

• module – a module of the order

EXAMPLES:

An ideal in an extension of a rational function field:

sage: K.<x> = FunctionField(QQ); R.<y> = K[]
sage: L.<y> = K.extension(y^2 - x^3 - 1)
sage: O = L.equation_order()
sage: I = O.ideal(y)
sage: I
Ideal (x^3 + 1, -y) of Order in Function field in y defined by y^2 - x^3 - 1
sage: I^2
Ideal (x^3 + 1, (-x^3 - 1)*y) of Order in Function field in y defined by y^2 - x^3 - 1

gen(i)

Return the i-th generator in the current basis of this ideal.

EXAMPLES:

sage: K.<x> = FunctionField(QQ); R.<y> = K[]
sage: L.<y> = K.extension(y^2 - x^3 - 1)
sage: O = L.equation_order()
sage: I = O.ideal(x^2 + 1)
sage: I.gen(1)
(x^2 + 1)*y

gens()

Return a set of generators of this ideal.

EXAMPLES:

sage: K.<x> = FunctionField(QQ); R.<y> = K[]
sage: L.<y> = K.extension(y^2 - x^3 - 1)
sage: O = L.equation_order()
sage: I = O.ideal(x^2 + 1)
sage: I.gens()
(x^2 + 1, (x^2 + 1)*y)

intersection(other)

Return the intersection of this ideal and other.

EXAMPLES:

sage: K.<x> = FunctionField(QQ); R.<y> = K[]
sage: L.<y> = K.extension(y^2 - x^3 - 1)
sage: O = L.equation_order()
sage: I = O.ideal(y^3); J = O.ideal(y^2)
sage: Z = I.intersection(J); Z
Ideal (x^6 + 2*x^3 + 1, (-x^3 - 1)*y) of Order in Function field in y defined by y^2 - x^3 - 1
sage: y^2 in Z
False
sage: y^3 in Z
True

module()

Return the module over the maximal order of the base field that underlies this ideal.

The formation of the module is compatible with the vector space corresponding to the function field.

OUTPUT:

• a module over the maximal order of the base field of the ideal

EXAMPLES:

sage: K.<x> = FunctionField(QQ); R.<y> = K[]
sage: L.<y> = K.extension(y^2 - x^3 - 1)
sage: O = L.equation_order();  O
Order in Function field in y defined by y^2 - x^3 - 1
sage: I = O.ideal(x^2 + 1)
sage: I.gens()
(x^2 + 1, (x^2 + 1)*y)
sage: I.module()
Free module of degree 2 and rank 2 over Maximal order of Rational function field in x over Rational Field
Echelon basis matrix:
[x^2 + 1       0]
[      0 x^2 + 1]
sage: V, from_V, to_V = L.vector_space(); V
Vector space of dimension 2 over Rational function field in x over Rational Field
sage: I.module().is_submodule(V)
True

ngens()

Return the number of generators in the basis.

EXAMPLES:

sage: K.<x> = FunctionField(QQ); R.<y> = K[]
sage: L.<y> = K.extension(y^2 - x^3 - 1)
sage: O = L.equation_order()
sage: I = O.ideal(x^2 + 1)
sage: I.ngens()
2

class sage.rings.function_field.ideal.FunctionFieldIdeal_polymod(ring, hnf, denominator=1)

Bases: sage.rings.function_field.ideal.FunctionFieldIdeal

Fractional ideals of algebraic function fields

INPUT:

• ring – order in a function field

• hnf – matrix in hermite normal form

• denominator – denominator

The rows of hnf is a basis of the ideal, which itself is denominator times the fractional ideal.

EXAMPLES:

sage: K.<x> = FunctionField(GF(2)); R.<y> = K[]
sage: L.<y> = K.extension(y^2 - x^3*y - x)
sage: O = L.maximal_order()
sage: O.ideal(y)
Ideal (y) of Maximal order of Function field in y defined by y^2 + x^3*y + x

basis_matrix()

Return the matrix of basis vectors of this ideal as a module.

The basis matrix is by definition the hermite norm form of the ideal divided by the denominator.

EXAMPLES:

sage: K.<x> = FunctionField(GF(2)); R.<t> = PolynomialRing(K)
sage: F.<y> = K.extension(t^3-x^2*(x^2+x+1)^2)
sage: O = F.maximal_order()
sage: I = O.ideal(x,1/y)
sage: I.denominator() * I.basis_matrix() == I.hnf()
True

denominator()

Return the denominator of this fractional ideal.

EXAMPLES:

sage: K.<x> = FunctionField(GF(7)); R.<y> = K[]
sage: L.<y> = K.extension(y^2 - x^3 - 1)
sage: O = L.maximal_order()
sage: I = O.ideal(y/(y+1))
sage: d = I.denominator(); d
x^3
sage: d in O
True

sage: K.<x> = FunctionField(QQ); R.<y> = K[]
sage: L.<y> = K.extension(y^2 - x^3 - 1)
sage: O = L.maximal_order()
sage: I = O.ideal(y/(y+1))
sage: d = I.denominator(); d
x^3
sage: d in O
True

gens()

Return a set of generators of this ideal.

This provides whatever set of generators as quickly as possible.

EXAMPLES:

sage: K.<x> = FunctionField(GF(2)); _.<Y> = K[]
sage: L.<y> = K.extension(Y^2 - x^3*Y - x)
sage: O = L.maximal_order()
sage: I = O.ideal(x+y)
sage: I.gens()
(x^4 + x^2 + x, y + x)

sage: L.<y> = K.extension(Y^2 +Y + x + 1/x)
sage: O = L.maximal_order()
sage: I = O.ideal(x+y)
sage: I.gens()
(x^3 + 1, y + x)

gens_over_base()

Return the generators of this ideal as a module over the maximal order of the base rational function field.

EXAMPLES:

sage: K.<x> = FunctionField(GF(2)); _.<Y> = K[]
sage: L.<y> = K.extension(Y^2 - x^3*Y - x)
sage: O = L.maximal_order()
sage: I = O.ideal(x+y)
sage: I.gens_over_base()
(x^4 + x^2 + x, y + x)

sage: L.<y> = K.extension(Y^2 + Y + x + 1/x)
sage: O = L.maximal_order()
sage: I = O.ideal(x+y)
sage: I.gens_over_base()
(x^3 + 1, y + x)

hnf()

Return the matrix in hermite normal form representing this ideal.

See also denominator()

EXAMPLES:

sage: K.<x> = FunctionField(GF(7)); R.<y> = K[]
sage: L.<y> = K.extension(y^2 - x^3 - 1)
sage: O = L.maximal_order()
sage: I = O.ideal(y*(y+1)); I.hnf()
[x^6 + x^3         0]
[  x^3 + 1         1]

sage: K.<x> = FunctionField(QQ); R.<y> = K[]
sage: L.<y> = K.extension(y^2 - x^3 - 1)
sage: O = L.maximal_order()
sage: I = O.ideal(y*(y+1)); I.hnf()
[x^6 + x^3         0]
[  x^3 + 1         1]

ideal_below()

Return the ideal below this ideal.

This is defined only for integral ideals.

EXAMPLES:

sage: K.<x> = FunctionField(GF(2)); _.<t> = K[]
sage: F.<y> = K.extension(t^3-x^2*(x^2+x+1)^2)
sage: O = F.maximal_order()
sage: I = O.ideal(x,1/y)
sage: I.ideal_below()
Traceback (most recent call last):
...
TypeError: not an integral ideal
sage: J = I.denominator() * I
sage: J.ideal_below()
Ideal (x^3 + x^2 + x) of Maximal order of Rational function field
in x over Finite Field of size 2

sage: K.<x> = FunctionField(GF(2)); _.<Y> = K[]
sage: L.<y> = K.extension(Y^2 + Y + x + 1/x)
sage: O = L.maximal_order()
sage: I = O.ideal(x,1/y)
sage: I.ideal_below()
Traceback (most recent call last):
...
TypeError: not an integral ideal
sage: J = I.denominator() * I
sage: J.ideal_below()
Ideal (x^3 + x) of Maximal order of Rational function field
in x over Finite Field of size 2

sage: K.<x> = FunctionField(QQ); _.<t> = K[]
sage: F.<y> = K.extension(t^3-x^2*(x^2+x+1)^2)
sage: O = F.maximal_order()
sage: I = O.ideal(x,1/y)
sage: I.ideal_below()
Traceback (most recent call last):
...
TypeError: not an integral ideal
sage: J = I.denominator() * I
sage: J.ideal_below()
Ideal (x^3 + x^2 + x) of Maximal order of Rational function field
in x over Rational Field

intersect(other)

Intersect this ideal with the other ideal as fractional ideals.

INPUT:

• other – ideal

EXAMPLES:

sage: K.<x> = FunctionField(GF(2)); _.<Y> = K[]
sage: L.<y> = K.extension(Y^2 - x^3*Y - x)
sage: O = L.maximal_order()
sage: I = O.ideal(x+y)
sage: J = O.ideal(x)
sage: I.intersect(J) == I * J * (I + J)^-1
True

is_integral()

Return True if this is an integral ideal.

EXAMPLES:

sage: K.<x> = FunctionField(GF(2)); _.<t> = PolynomialRing(K)
sage: F.<y> = K.extension(t^3-x^2*(x^2+x+1)^2)
sage: O = F.maximal_order()
sage: I = O.ideal(x,1/y)
sage: I.is_integral()
False
sage: J = I.denominator() * I
sage: J.is_integral()
True

sage: K.<x> = FunctionField(GF(2)); _.<Y> = K[]
sage: L.<y> = K.extension(Y^2 + Y + x + 1/x)
sage: O = L.maximal_order()
sage: I = O.ideal(x,1/y)
sage: I.is_integral()
False
sage: J = I.denominator() * I
sage: J.is_integral()
True

sage: K.<x> = FunctionField(QQ); _.<t> = PolynomialRing(K)
sage: F.<y> = K.extension(t^3-x^2*(x^2+x+1)^2)
sage: O = F.maximal_order()
sage: I = O.ideal(x,1/y)
sage: I.is_integral()
False
sage: J = I.denominator() * I
sage: J.is_integral()
True

is_prime()

Return True if this ideal is a prime ideal.

EXAMPLES:

sage: K.<x> = FunctionField(GF(2)); _.<t> = PolynomialRing(K)
sage: F.<y> = K.extension(t^3-x^2*(x^2+x+1)^2)
sage: O = F.maximal_order()
sage: I = O.ideal(y)
sage: [f.is_prime() for f,_ in I.factor()]
[True, True]

sage: K.<x> = FunctionField(GF(2)); _.<Y> = K[]
sage: L.<y> = K.extension(Y^2 + Y + x + 1/x)
sage: O = L.maximal_order()
sage: I = O.ideal(y)
sage: [f.is_prime() for f,_ in I.factor()]
[True, True]

sage: K.<x> = FunctionField(QQ); _.<t> = PolynomialRing(K)
sage: F.<y> = K.extension(t^3-x^2*(x^2+x+1)^2)
sage: O = F.maximal_order()
sage: I = O.ideal(y)
sage: [f.is_prime() for f,_ in I.factor()]
[True, True]

module()

Return the module, that is the ideal viewed as a module over the base maximal order.

EXAMPLES:

sage: K.<x> = FunctionField(GF(7)); R.<y> = K[]
sage: F.<y> = K.extension(y^2 - x^3 - 1)
sage: O = F.maximal_order()
sage: I = O.ideal(x,1/y)
sage: I.module()
Free module of degree 2 and rank 2 over Maximal order
of Rational function field in x over Finite Field of size 7
Echelon basis matrix:
[          1           0]
[          0 1/(x^3 + 1)]

norm()

Return the norm of this fractional ideal.

EXAMPLES:

sage: K.<x> = FunctionField(GF(2)); _.<t> = PolynomialRing(K)
sage: F.<y> = K.extension(t^3-x^2*(x^2+x+1)^2)
sage: O = F.maximal_order()
sage: i1 = O.ideal(x)
sage: i2 = O.ideal(y)
sage: i3 = i1 * i2
sage: i3.norm() == i1.norm() * i2.norm()
True
sage: i1.norm()
x^3
sage: i1.norm() == x ** F.degree()
True
sage: i2.norm()
x^6 + x^4 + x^2
sage: i2.norm() == y.norm()
True

sage: K.<x> = FunctionField(GF(2)); _.<Y> = K[]
sage: L.<y> = K.extension(Y^2 + Y + x + 1/x)
sage: O = L.maximal_order()
sage: i1 = O.ideal(x)
sage: i2 = O.ideal(y)
sage: i3 = i1 * i2
sage: i3.norm() == i1.norm() * i2.norm()
True
sage: i1.norm()
x^2
sage: i1.norm() == x ** L.degree()
True
sage: i2.norm()
(x^2 + 1)/x
sage: i2.norm() == y.norm()
True

prime_below()

Return the prime lying below this prime ideal.

EXAMPLES:

sage: K.<x> = FunctionField(GF(2)); _.<Y> = K[]
sage: F.<y> = K.extension(Y^3 - x^2*(x^2 + x + 1)^2)
sage: O = F.maximal_order()
sage: I = O.ideal(y)
sage: [f.prime_below() for f,_ in I.factor()]
[Ideal (x) of Maximal order of Rational function field in x
over Finite Field of size 2, Ideal (x^2 + x + 1) of Maximal order
of Rational function field in x over Finite Field of size 2]

sage: K.<x> = FunctionField(GF(2)); _.<Y> = K[]
sage: L.<y> = K.extension(Y^2 + Y + x + 1/x)
sage: O = L.maximal_order()
sage: I = O.ideal(y)
sage: [f.prime_below() for f,_ in I.factor()]
[Ideal (x) of Maximal order of Rational function field in x over Finite Field of size 2,
 Ideal (x + 1) of Maximal order of Rational function field in x over Finite Field of size 2]

sage: K.<x> = FunctionField(QQ); _.<Y> = K[]
sage: F.<y> = K.extension(Y^3 - x^2*(x^2 + x + 1)^2)
sage: O = F.maximal_order()
sage: I = O.ideal(y)
sage: [f.prime_below() for f,_ in I.factor()]
[Ideal (x) of Maximal order of Rational function field in x over Rational Field,
 Ideal (x^2 + x + 1) of Maximal order of Rational function field in x over Rational Field]

valuation(ideal)

Return the valuation of ideal at this prime ideal.

INPUT:

• ideal – fractional ideal

EXAMPLES:

sage: K.<x> = FunctionField(GF(2)); _.<t> = K[]
sage: F.<y> = K.extension(t^3 - x^2*(x^2 + x + 1)^2)
sage: O = F.maximal_order()
sage: I = O.ideal(x, (1/(x^3 + x^2 + x))*y^2)
sage: I.is_prime()
True
sage: J = O.ideal(y)
sage: I.valuation(J)
2

sage: K.<x> = FunctionField(GF(2)); _.<Y> = K[]
sage: L.<y> = K.extension(Y^2 + Y + x + 1/x)
sage: O = L.maximal_order()
sage: I = O.ideal(y)
sage: [f.valuation(I) for f,_ in I.factor()]
[-1, 2]

The method closely follows Algorithm 4.8.17 of [Coh1993].

class sage.rings.function_field.ideal.FunctionFieldIdeal_rational(ring, gen)

Bases: sage.rings.function_field.ideal.FunctionFieldIdeal

Fractional ideals of the maximal order of a rational function field.

INPUT:

• ring – the maximal order of the rational function field.

• gen – generator of the ideal, an element of the function field.

EXAMPLES:

sage: K.<x> = FunctionField(QQ)
sage: O = K.maximal_order()
sage: I = O.ideal(1/(x^2+x)); I
Ideal (1/(x^2 + x)) of Maximal order of Rational function field in x over Rational Field

denominator()

Return the denominator of this fractional ideal.

EXAMPLES:

sage: F.<x> = FunctionField(QQ)
sage: O = F.maximal_order()
sage: I = O.ideal(x/(x^2+1))
sage: I.denominator()
x^2 + 1

gen()

Return the unique generator of this ideal.

EXAMPLES:

sage: K.<x> = FunctionField(GF(4))
sage: O = K.maximal_order()
sage: I = O.ideal(x^2+x)
sage: I.gen()
x^2 + x

gens()

Return the tuple of the unique generator of this ideal.

EXAMPLES:

sage: K.<x> = FunctionField(GF(4))
sage: O = K.maximal_order()
sage: I = O.ideal(x^2+x)
sage: I.gens()
(x^2 + x,)

gens_over_base()

Return the generator of this ideal as a rank one module over the maximal order.

EXAMPLES:

sage: K.<x> = FunctionField(GF(4))
sage: O = K.maximal_order()
sage: I = O.ideal(x^2+x)
sage: I.gens_over_base()
(x^2 + x,)

is_prime()

Return True if this is a prime ideal.

EXAMPLES:

sage: K.<x> = FunctionField(QQ)
sage: O = K.maximal_order()
sage: I = O.ideal(x^3+x^2)
sage: [f.is_prime() for f,m in I.factor()]
[True, True]

module()

Return the module, that is the ideal viewed as a module over the ring.

EXAMPLES:

sage: K.<x> = FunctionField(QQ)
sage: O = K.maximal_order()
sage: I = O.ideal(x^3+x^2)
sage: I.module()
Free module of degree 1 and rank 1 over Maximal order of Rational
function field in x over Rational Field
Echelon basis matrix:
[x^3 + x^2]
sage: J = 0*I
sage: J.module()
Free module of degree 1 and rank 0 over Maximal order of Rational
function field in x over Rational Field
Echelon basis matrix:
[]

valuation(ideal)

Return the valuation of the ideal at this prime ideal.

INPUT:

• ideal – fractional ideal

EXAMPLES:

sage: F.<x> = FunctionField(QQ)
sage: O = F.maximal_order()
sage: I = O.ideal(x^2*(x^2+x+1)^3)
sage: [f.valuation(I) for f,_ in I.factor()]
[2, 3]

class sage.rings.function_field.ideal.IdealMonoid(R)

Bases: sage.structure.unique_representation.UniqueRepresentation, sage.structure.parent.Parent

The monoid of ideals in orders of function fields.

INPUT:

• R – order

EXAMPLES:

sage: K.<x> = FunctionField(GF(2))
sage: O = K.maximal_order()
sage: M = O.ideal_monoid(); M
Monoid of ideals of Maximal order of Rational function field in x over Finite Field of size 2

ring()

Return the ring of which this is the ideal monoid.

EXAMPLES:

sage: K.<x> = FunctionField(GF(2))
sage: O = K.maximal_order()
sage: M = O.ideal_monoid(); M.ring() is O
True