def ideal_with_gens_over_base(self, gens): """ Returns the fractional ideal with basis ``gens`` over the maximal order of the base field. That this is really an ideal is not checked. INPUT: - ``gens`` -- list of elements that are a basis for the ideal over the maximal order of the base field EXAMPLES: We construct an ideal in a rational function field:: sage: K.<y> = FunctionField(QQ) sage: O = K.maximal_order() sage: I = O.ideal_with_gens_over_base([y]); I Ideal (y) of Maximal order in Rational function field in y over Rational Field sage: I*I Ideal (y^2) of Maximal order in Rational function field in y over Rational Field We construct some ideals in a nontrivial function field:: sage: K.<x> = FunctionField(GF(7)); 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 + 6*x^3 + 6 sage: I = O.ideal_with_gens_over_base([1, y]); I Ideal (1, y) of Order in Function field in y defined by y^2 + 6*x^3 + 6 sage: I.module() Free module of degree 2 and rank 2 over Maximal order in Rational function field in x over Finite Field of size 7 Echelon basis matrix: [1 0] [0 1] There is no check if the resulting object is really an ideal:: 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_with_gens_over_base([y]); I Ideal (y) of Order in Function field in y defined by y^2 + 6*x^3 + 6 sage: y in I True sage: y^2 in I False """ from function_field_ideal import ideal_with_gens_over_base return ideal_with_gens_over_base(self, [self(a) for a in gens])
def ideal_with_gens_over_base(self, gens): """ Return the fractional ideal with given generators over the maximal ideal of the base field. That this is really an ideal is not checked. INPUT: - ``basis`` -- list of elements that are a basis for the ideal over the maximal order of the base field EXAMPLES:: We construct an ideal in a rational function field:: sage: R.<y> = FunctionField(QQ) sage: S = R.maximal_order() sage: I = S.ideal_with_gens_over_base([y]); I Ideal (y) of Maximal order in Rational function field in y over Rational Field sage: I*I Ideal (y^2) of Maximal order in Rational function field in y over Rational Field We construct some ideals in a nontrivial function field:: sage: R.<x> = FunctionField(GF(7)); S.<y> = R[] sage: L.<y> = R.extension(y^2 - x^3 - 1) sage: M = L.equation_order(); M Order in Function field in y defined by y^2 + 6*x^3 + 6 sage: I = M.ideal_with_gens_over_base([1, y]); I Ideal (1, y) of Order in Function field in y defined by y^2 + 6*x^3 + 6 sage: I.module() Free module of degree 2 and rank 2 over Maximal order in Rational function field in x over Finite Field of size 7 Echelon basis matrix: [1 0] [0 1] """ from function_field_ideal import ideal_with_gens_over_base return ideal_with_gens_over_base(self, [self(a) for a in gens])