# Ideals in Function Fields¶

AUTHORS:

• William Stein (2010): initial version
• Maarten Derickx (2011-09-14): fixed ideal_with_gens_over_base()

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 in Rational function field in x over Rational Field
sage: I^2
Ideal (x^6 + 2*x^3 + 1) of Maximal order in Rational function field in x over Rational Field
sage: ~I
Ideal (1/(x^3 + 1)) of Maximal order in Rational function field in x over Rational Field
sage: ~I * I
Ideal (1) of Maximal order in 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
sage: ~I
Ideal (-1, (1/(x^3 + 1))*y) of Order in Function field in y defined by y^2 - x^3 - 1
sage: ~I * I
Ideal (1, y) of Order in Function field in y defined by y^2 - x^3 - 1
sage: I.intersection(~I)
Ideal (x^3 + 1, -y) of Order in Function field in y defined by y^2 - x^3 - 1

class sage.rings.function_field.function_field_ideal.FunctionFieldIdeal(ring, gens, coerce=True)

A fractional ideal of a function field.

EXAMPLES:

sage: K.<x> = FunctionField(GF(7))
sage: O = K.maximal_order()
sage: I = O.ideal(x^3+1)
sage: isinstance(I, sage.rings.function_field.function_field_ideal.FunctionFieldIdeal)
True

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

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

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

intersection(other)

Return the intersection of the ideals self and other.

EXAMPLES:

sage: K.<x> = FunctionField(GF(7)); 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, (6*x^3 + 6)*y) of Order in Function field in y defined by y^2 + 6*x^3 + 6
sage: y^2 in Z
False
sage: y^3 in Z
True

module()

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

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

OUTPUT:

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

EXAMPLES:

sage: K.<x> = FunctionField(GF(7))
sage: O = K.maximal_order(); O
Maximal order in 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_with_gens_over_base([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 in Rational function field in x over Finite Field of size 7
User 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

sage.rings.function_field.function_field_ideal.ideal_with_gens(R, gens)

Return fractional ideal in the order R with generators gens over R.

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: sage.rings.function_field.function_field_ideal.ideal_with_gens(O, [y])
Ideal (x^3 + 1, -y) of Order in Function field in y defined by y^2 - x^3 - 1

sage.rings.function_field.function_field_ideal.ideal_with_gens_over_base(R, gens)

Return fractional ideal in the order R with generators gens over the maximal order of the base field.

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: sage.rings.function_field.function_field_ideal.ideal_with_gens_over_base(O, [x^3+1,-y])
Ideal (x^3 + 1, -y) of Order in Function field in y defined by y^2 - x^3 - 1


TESTS:

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


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