Places of function fields: rational#
- class sage.rings.function_field.place_rational.FunctionFieldPlace_rational(parent, prime)#
Bases:
FunctionFieldPlace
Places of rational function fields.
- degree()#
Return the degree of the place.
EXAMPLES:
sage: F.<x> = FunctionField(GF(2)) sage: O = F.maximal_order() sage: i = O.ideal(x^2 + x + 1) sage: p = i.place() sage: p.degree() 2
- is_infinite_place()#
Return
True
if the place is at infinite.EXAMPLES:
sage: F.<x> = FunctionField(GF(2)) sage: F.places() [Place (1/x), Place (x), Place (x + 1)] sage: [p.is_infinite_place() for p in F.places()] [True, False, False]
- local_uniformizer()#
Return a local uniformizer of the place.
EXAMPLES:
sage: F.<x> = FunctionField(GF(2)) sage: F.places() [Place (1/x), Place (x), Place (x + 1)] sage: [p.local_uniformizer() for p in F.places()] [1/x, x, x + 1]
- residue_field(name=None)#
Return the residue field of the place.
EXAMPLES:
sage: F.<x> = FunctionField(GF(2)) sage: O = F.maximal_order() sage: p = O.ideal(x^2 + x + 1).place() sage: k, fr_k, to_k = p.residue_field() # needs sage.rings.function_field sage: k # needs sage.rings.function_field Finite Field in z2 of size 2^2 sage: fr_k # needs sage.rings.function_field Ring morphism: From: Finite Field in z2 of size 2^2 To: Valuation ring at Place (x^2 + x + 1) sage: to_k # needs sage.rings.function_field Ring morphism: From: Valuation ring at Place (x^2 + x + 1) To: Finite Field in z2 of size 2^2
- valuation_ring()#
Return the valuation ring at the place.
EXAMPLES:
sage: K.<x> = FunctionField(GF(2)); _.<Y> = K[] sage: L.<y> = K.extension(Y^2 + Y + x + 1/x) # needs sage.rings.function_field sage: p = L.places_finite()[0] # needs sage.rings.function_field sage: p.valuation_ring() # needs sage.rings.function_field Valuation ring at Place (x, x*y)