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# 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)

  Bases: `sage.rings.ideal.Ideal_generic`

  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)

  Bases: `sage.rings.function_field.function_field_ideal.FunctionFieldIdeal`

  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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