Orders of function fields: rational¶
- class sage.rings.function_field.order_rational.FunctionFieldMaximalOrderInfinite_rational(field, category=None)[source]¶
Bases:
FunctionFieldMaximalOrderInfiniteMaximal infinite orders of rational function fields.
INPUT:
field– a rational function field
EXAMPLES:
sage: K.<t> = FunctionField(GF(19)); K Rational function field in t over Finite Field of size 19 sage: R = K.maximal_order_infinite(); R Maximal infinite order of Rational function field in t over Finite Field of size 19
>>> from sage.all import * >>> K = FunctionField(GF(Integer(19)), names=('t',)); (t,) = K._first_ngens(1); K Rational function field in t over Finite Field of size 19 >>> R = K.maximal_order_infinite(); R Maximal infinite order of Rational function field in t over Finite Field of size 19
- basis()[source]¶
Return the basis (=1) of the order as a module over the polynomial ring.
EXAMPLES:
sage: K.<t> = FunctionField(GF(19)) sage: O = K.maximal_order() sage: O.basis() (1,)
>>> from sage.all import * >>> K = FunctionField(GF(Integer(19)), names=('t',)); (t,) = K._first_ngens(1) >>> O = K.maximal_order() >>> O.basis() (1,)
- gen(n=0)[source]¶
Return the \(n\)-th generator of
self. Since there is only one generator \(n\) must be \(0\).EXAMPLES:
sage: O = FunctionField(QQ,'y').maximal_order() sage: O.gen() y sage: O.gen(1) Traceback (most recent call last): ... IndexError: there is only one generator
>>> from sage.all import * >>> O = FunctionField(QQ,'y').maximal_order() >>> O.gen() y >>> O.gen(Integer(1)) Traceback (most recent call last): ... IndexError: there is only one generator
- ideal(*gens)[source]¶
Return the fractional ideal generated by
gens.INPUT:
gens– elements of the function field
EXAMPLES:
sage: K.<x> = FunctionField(QQ) sage: O = K.maximal_order_infinite() sage: O.ideal(x) Ideal (x) of Maximal infinite order of 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 (x^3) of Maximal infinite order of 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^5) of Maximal infinite order of Rational function field in x over Rational Field sage: O.ideal(I) Ideal (x^5) of Maximal infinite order of Rational function field in x over Rational Field
>>> from sage.all import * >>> K = FunctionField(QQ, names=('x',)); (x,) = K._first_ngens(1) >>> O = K.maximal_order_infinite() >>> O.ideal(x) Ideal (x) of Maximal infinite order of Rational function field in x over Rational Field >>> O.ideal([x, Integer(1)/x]) == O.ideal(x ,Integer(1)/x) # multiple generators may be given as a list True >>> O.ideal(x**Integer(3) + Integer(1), x**Integer(3) + Integer(6)) Ideal (x^3) of Maximal infinite order of Rational function field in x over Rational Field >>> I = O.ideal((x**Integer(2)+Integer(1))*(x**Integer(3)+Integer(1)), (x**Integer(3)+Integer(6))*(x**Integer(2)+Integer(1))); I Ideal (x^5) of Maximal infinite order of Rational function field in x over Rational Field >>> O.ideal(I) Ideal (x^5) of Maximal infinite order of Rational function field in x over Rational Field
- ngens()[source]¶
Return 1 the number of generators of the order.
EXAMPLES:
sage: FunctionField(QQ,'y').maximal_order().ngens() 1
>>> from sage.all import * >>> FunctionField(QQ,'y').maximal_order().ngens() 1
- prime_ideal()[source]¶
Return the unique prime ideal of the order.
EXAMPLES:
sage: K.<t> = FunctionField(GF(19)) sage: O = K.maximal_order_infinite() sage: O.prime_ideal() Ideal (1/t) of Maximal infinite order of Rational function field in t over Finite Field of size 19
>>> from sage.all import * >>> K = FunctionField(GF(Integer(19)), names=('t',)); (t,) = K._first_ngens(1) >>> O = K.maximal_order_infinite() >>> O.prime_ideal() Ideal (1/t) of Maximal infinite order of Rational function field in t over Finite Field of size 19
- class sage.rings.function_field.order_rational.FunctionFieldMaximalOrder_rational(field)[source]¶
Bases:
FunctionFieldMaximalOrderMaximal orders of rational function fields.
INPUT:
field– a function field
EXAMPLES:
sage: K.<t> = FunctionField(GF(19)); K Rational function field in t over Finite Field of size 19 sage: R = K.maximal_order(); R Maximal order of Rational function field in t over Finite Field of size 19
>>> from sage.all import * >>> K = FunctionField(GF(Integer(19)), names=('t',)); (t,) = K._first_ngens(1); K Rational function field in t over Finite Field of size 19 >>> R = K.maximal_order(); R Maximal order of Rational function field in t over Finite Field of size 19
- basis()[source]¶
Return the basis (=1) of the order as a module over the polynomial ring.
EXAMPLES:
sage: K.<t> = FunctionField(GF(19)) sage: O = K.maximal_order() sage: O.basis() (1,)
>>> from sage.all import * >>> K = FunctionField(GF(Integer(19)), names=('t',)); (t,) = K._first_ngens(1) >>> O = K.maximal_order() >>> O.basis() (1,)
- gen(n=0)[source]¶
Return the
n-th generator of the order. Since there is only one generatornmust be 0.EXAMPLES:
sage: O = FunctionField(QQ,'y').maximal_order() sage: O.gen() y sage: O.gen(1) Traceback (most recent call last): ... IndexError: there is only one generator
>>> from sage.all import * >>> O = FunctionField(QQ,'y').maximal_order() >>> O.gen() y >>> O.gen(Integer(1)) Traceback (most recent call last): ... IndexError: there is only one generator
- ideal(*gens)[source]¶
Return the fractional ideal generated by
gens.INPUT:
gens– elements of the function field
EXAMPLES:
sage: K.<x> = FunctionField(QQ) sage: O = K.maximal_order() sage: O.ideal(x) Ideal (x) of Maximal order of 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 of 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 of Rational function field in x over Rational Field sage: O.ideal(I) Ideal (x^2 + 1) of Maximal order of Rational function field in x over Rational Field
>>> from sage.all import * >>> K = FunctionField(QQ, names=('x',)); (x,) = K._first_ngens(1) >>> O = K.maximal_order() >>> O.ideal(x) Ideal (x) of Maximal order of Rational function field in x over Rational Field >>> O.ideal([x, Integer(1)/x]) == O.ideal(x, Integer(1)/x) # multiple generators may be given as a list True >>> O.ideal(x**Integer(3) + Integer(1), x**Integer(3) + Integer(6)) Ideal (1) of Maximal order of Rational function field in x over Rational Field >>> I = O.ideal((x**Integer(2)+Integer(1))*(x**Integer(3)+Integer(1)), (x**Integer(3)+Integer(6))*(x**Integer(2)+Integer(1))); I Ideal (x^2 + 1) of Maximal order of Rational function field in x over Rational Field >>> O.ideal(I) Ideal (x^2 + 1) of Maximal order of Rational function field in x over Rational Field
- ideal_with_gens_over_base(gens)[source]¶
Return the fractional ideal with generators
gens.INPUT:
gens– elements of the function field
EXAMPLES:
sage: K.<x> = FunctionField(QQ); R.<y> = K[] sage: L.<y> = K.extension(y^2 - x^3 - 1) # needs sage.rings.function_field sage: O = L.equation_order() # needs sage.rings.function_field sage: O.ideal_with_gens_over_base([x^3 + 1, -y]) # needs sage.rings.function_field Ideal (x^3 + 1, -y) of Order in Function field in y defined by y^2 - x^3 - 1
>>> from sage.all import * >>> K = FunctionField(QQ, names=('x',)); (x,) = K._first_ngens(1); R = K['y']; (y,) = R._first_ngens(1) >>> L = K.extension(y**Integer(2) - x**Integer(3) - Integer(1), names=('y',)); (y,) = L._first_ngens(1)# needs sage.rings.function_field >>> O = L.equation_order() # needs sage.rings.function_field >>> O.ideal_with_gens_over_base([x**Integer(3) + Integer(1), -y]) # needs sage.rings.function_field Ideal (x^3 + 1, -y) of Order in Function field in y defined by y^2 - x^3 - 1