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):
"""
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])
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])
