Ideals of function fields: rational#
- class sage.rings.function_field.ideal_rational.FunctionFieldIdealInfinite_rational(ring, gen)#
Bases:
FunctionFieldIdealInfinite
Fractional ideal of the maximal order of rational function field.
INPUT:
ring
– infinite maximal ordergen
– generator
Note that the infinite maximal order is a principal ideal domain.
EXAMPLES:
sage: K.<x> = FunctionField(GF(2)) sage: Oinf = K.maximal_order_infinite() sage: Oinf.ideal(x) Ideal (x) of Maximal infinite order of Rational function field in x over Finite Field of size 2
- gen()#
Return the generator of this principal ideal.
EXAMPLES:
sage: # needs sage.rings.finite_rings sage: K.<x> = FunctionField(GF(2)) sage: Oinf = K.maximal_order_infinite() sage: I = Oinf.ideal((x+1)/(x^3+x), (x^2+1)/x^4) sage: I.gen() 1/x^2
- gens()#
Return the generator of this principal ideal.
EXAMPLES:
sage: # needs sage.rings.finite_rings sage: K.<x> = FunctionField(GF(2)) sage: Oinf = K.maximal_order_infinite() sage: I = Oinf.ideal((x+1)/(x^3+x), (x^2+1)/x^4) sage: I.gens() (1/x^2,)
- gens_over_base()#
Return the generator of this ideal as a rank one module over the infinite maximal order.
EXAMPLES:
sage: # needs sage.rings.finite_rings sage: K.<x> = FunctionField(GF(2)) sage: Oinf = K.maximal_order_infinite() sage: I = Oinf.ideal((x+1)/(x^3+x), (x^2+1)/x^4) sage: I.gens_over_base() (1/x^2,)
- is_prime()#
Return
True
if this ideal is a prime ideal.EXAMPLES:
sage: # needs sage.rings.finite_rings sage: K.<x> = FunctionField(GF(2)) sage: Oinf = K.maximal_order_infinite() sage: I = Oinf.ideal(x/(x^2 + 1)) sage: I.is_prime() True
- valuation(ideal)#
Return the valuation of
ideal
at this prime ideal.INPUT:
ideal
– fractional ideal
EXAMPLES:
sage: F.<x> = FunctionField(QQ) sage: O = F.maximal_order_infinite() sage: p = O.ideal(1/x) sage: p.valuation(O.ideal(x/(x+1))) 0 sage: p.valuation(O.ideal(0)) +Infinity
- class sage.rings.function_field.ideal_rational.FunctionFieldIdeal_rational(ring, gen)#
Bases:
FunctionFieldIdeal
Fractional ideals of the maximal order of a rational function field.
INPUT:
ring
– the maximal order of the rational function field.gen
– generator of the ideal, an element of the function field.
EXAMPLES:
sage: K.<x> = FunctionField(QQ) sage: O = K.maximal_order() sage: I = O.ideal(1/(x^2+x)); I Ideal (1/(x^2 + x)) of Maximal order of Rational function field in x over Rational Field
- denominator()#
Return the denominator of this fractional ideal.
EXAMPLES:
sage: F.<x> = FunctionField(QQ) sage: O = F.maximal_order() sage: I = O.ideal(x/(x^2+1)) sage: I.denominator() x^2 + 1
- gen()#
Return the unique generator of this ideal.
EXAMPLES:
sage: # needs sage.rings.finite_rings sage: K.<x> = FunctionField(GF(4)) sage: O = K.maximal_order() sage: I = O.ideal(x^2 + x) sage: I.gen() x^2 + x
- gens()#
Return the tuple of the unique generator of this ideal.
EXAMPLES:
sage: # needs sage.rings.finite_rings sage: K.<x> = FunctionField(GF(4)) sage: O = K.maximal_order() sage: I = O.ideal(x^2 + x) sage: I.gens() (x^2 + x,)
- gens_over_base()#
Return the generator of this ideal as a rank one module over the maximal order.
EXAMPLES:
sage: # needs sage.rings.finite_rings sage: K.<x> = FunctionField(GF(4)) sage: O = K.maximal_order() sage: I = O.ideal(x^2 + x) sage: I.gens_over_base() (x^2 + x,)
- is_prime()#
Return
True
if this is a prime ideal.EXAMPLES:
sage: K.<x> = FunctionField(QQ) sage: O = K.maximal_order() sage: I = O.ideal(x^3 + x^2) sage: [f.is_prime() for f,m in I.factor()] # needs sage.libs.pari [True, True]
- module()#
Return the module, that is the ideal viewed as a module over the ring.
EXAMPLES:
sage: K.<x> = FunctionField(QQ) sage: O = K.maximal_order() sage: I = O.ideal(x^3 + x^2) sage: I.module() # needs sage.modules Free module of degree 1 and rank 1 over Maximal order of Rational function field in x over Rational Field Echelon basis matrix: [x^3 + x^2] sage: J = 0*I sage: J.module() # needs sage.modules Free module of degree 1 and rank 0 over Maximal order of Rational function field in x over Rational Field Echelon basis matrix: []
- valuation(ideal)#
Return the valuation of the ideal at this prime ideal.
INPUT:
ideal
– fractional ideal
EXAMPLES:
sage: F.<x> = FunctionField(QQ) sage: O = F.maximal_order() sage: I = O.ideal(x^2*(x^2+x+1)^3) sage: [f.valuation(I) for f,_ in I.factor()] # needs sage.libs.pari [2, 3]