def ideal(self, *gens): """ Returns the fractional ideal generated by ``gens``. EXAMPLES:: sage: K.<x> = FunctionField(QQ) sage: O = K.maximal_order() sage: O.ideal(x) Ideal (x) of Maximal order in Rational function field in x over Rational Field sage: O.ideal([x,1/x]) == O.ideal(x,1/x) # multiple generators may be given as a list True sage: O.ideal(x^3+1,x^3+6) Ideal (1) of Maximal order in Rational function field in x over Rational Field sage: I = O.ideal((x^2+1)*(x^3+1),(x^3+6)*(x^2+1)); I Ideal (x^2 + 1) of Maximal order in Rational function field in x over Rational Field sage: O.ideal(I) Ideal (x^2 + 1) of Maximal order in Rational function field in x over Rational Field """ if len(gens) == 1: gens = gens[0] if not isinstance(gens, (list, tuple)): if is_Ideal(gens): gens = gens.gens() else: gens = (gens, ) from function_field_ideal import ideal_with_gens return ideal_with_gens(self, gens)
def ideal(self, *gens): """ Returns the fractional ideal generated by ``gens``. EXAMPLES:: sage: K.<x> = FunctionField(QQ) sage: O = K.maximal_order() sage: O.ideal(x) Ideal (x) of Maximal order in Rational function field in x over Rational Field sage: O.ideal([x,1/x]) == O.ideal(x,1/x) # multiple generators may be given as a list True sage: O.ideal(x^3+1,x^3+6) Ideal (1) of Maximal order in Rational function field in x over Rational Field sage: I = O.ideal((x^2+1)*(x^3+1),(x^3+6)*(x^2+1)); I Ideal (x^2 + 1) of Maximal order in Rational function field in x over Rational Field sage: O.ideal(I) Ideal (x^2 + 1) of Maximal order in Rational function field in x over Rational Field """ if len(gens) == 1: gens = gens[0] if not isinstance(gens, (list, tuple)): if is_Ideal(gens): gens = gens.gens() else: gens = (gens,) from function_field_ideal import ideal_with_gens return ideal_with_gens(self, gens)
def ideal(self, *gens): """ Return the fractional ideal generated by the element gens or the elements in gens if gens is a list. EXAMPLES:: sage: R.<y> = FunctionField(QQ) sage: S = R.maximal_order() sage: S.ideal(y) Ideal (y) of Maximal order in Rational function field in y over Rational Field A fractional ideal of a nontrivial extension:: sage: R.<x> = FunctionField(GF(7)); S.<y> = R[] sage: L.<y> = R.extension(y^2 - x^3 - 1) sage: M = L.equation_order() sage: M.ideal(1/y) Ideal (1, (6/(x^3 + 1))*y) of Order in Function field in y defined by y^2 + 6*x^3 + 6 """ if len(gens) == 1: gens = gens[0] if not isinstance(gens, (list, tuple)): gens = [gens] from function_field_ideal import ideal_with_gens return ideal_with_gens(self, gens)
def ideal(self, *gens): """ Returns the fractional ideal generated by the elements in ``gens``. INPUT: - ``gens`` -- a list of generators or an ideal in a ring which coerces to this order. EXAMPLES:: sage: K.<y> = FunctionField(QQ) sage: O = K.maximal_order() sage: O.ideal(y) Ideal (y) of Maximal order in Rational function field in y over Rational Field sage: O.ideal([y,1/y]) == O.ideal(y,1/y) # multiple generators may be given as a list True A fractional ideal of a nontrivial extension:: sage: K.<x> = FunctionField(GF(7)); R.<y> = K[] sage: O = K.maximal_order() sage: I = O.ideal(x^2-4) sage: L.<y> = K.extension(y^2 - x^3 - 1) sage: S = L.equation_order() sage: S.ideal(1/y) Ideal (1, (6/(x^3 + 1))*y) of Order in Function field in y defined by y^2 + 6*x^3 + 6 sage: I2 = S.ideal(x^2-4); I2 Ideal (x^2 + 3, (x^2 + 3)*y) of Order in Function field in y defined by y^2 + 6*x^3 + 6 sage: I2 == S.ideal(I) True """ if len(gens) == 1: gens = gens[0] if not isinstance(gens, (list, tuple)): if is_Ideal(gens): gens = gens.gens() else: gens = [gens] from function_field_ideal import ideal_with_gens return ideal_with_gens(self, gens)