Valuation rings of function fields¶
A valuation ring of a function field is associated with a place of the function field. The valuation ring consists of all elements of the function field that have nonnegative valuation at the place.
EXAMPLES:
sage: K.<x> = FunctionField(GF(2)); _.<Y> = K[]
sage: L.<y> = K.extension(Y^2 + Y + x + 1/x)
sage: p = L.places_finite()[0]
sage: p
Place (x, x*y)
sage: R = p.valuation_ring()
sage: R
Valuation ring at Place (x, x*y)
sage: R.place() == p
True
Thus any nonzero element or its inverse of the function field lies in the valuation ring, as shown in the following example:
sage: f = y/(1+y)
sage: f in R
True
sage: f not in R
False
sage: f.valuation(p)
0
The residue field at the place is defined as the quotient ring of the valuation
ring modulo its unique maximal ideal. The method residue_field()
of the
valuation ring returns an extension field of the constant base field, isomorphic
to the residue field, along with lifting and evaluation homomorphisms:
sage: k,phi,psi = R.residue_field()
sage: k
Finite Field of size 2
sage: phi
Ring morphism:
From: Finite Field of size 2
To: Valuation ring at Place (x, x*y)
sage: psi
Ring morphism:
From: Valuation ring at Place (x, x*y)
To: Finite Field of size 2
sage: psi(f) in k
True
AUTHORS:
Kwankyu Lee (2017-04-30): initial version
- class sage.rings.function_field.valuation_ring.FunctionFieldValuationRing(field, place, category=None)¶
Bases:
sage.structure.unique_representation.UniqueRepresentation
,sage.structure.parent.Parent
Base class for valuation rings of function fields.
INPUT:
field
– function fieldplace
– place of the function field
EXAMPLES:
sage: K.<x> = FunctionField(GF(2)); _.<Y> = K[] sage: L.<y> = K.extension(Y^2 + Y + x + 1/x) sage: p = L.places_finite()[0] sage: p.valuation_ring() Valuation ring at Place (x, x*y)
- place()¶
Return the place associated with the valuation ring.
EXAMPLES:
sage: K.<x> = FunctionField(GF(2)); _.<Y> = K[] sage: L.<y> = K.extension(Y^2 + Y + x + 1/x) sage: p = L.places_finite()[0] sage: R = p.valuation_ring() sage: p == R.place() True
- residue_field(name=None)¶
Return the residue field of the valuation ring together with the maps from and to it.
INPUT:
name
– string; name of the generator of the field
OUTPUT:
a field isomorphic to the residue field
a ring homomorphism from the valuation ring to the field
a ring homomorphism from the field to the valuation ring
EXAMPLES:
sage: K.<x> = FunctionField(GF(2)); _.<Y> = K[] sage: L.<y> = K.extension(Y^2 + Y + x + 1/x) sage: p = L.places_finite()[0] sage: R = p.valuation_ring() sage: k, fr_k, to_k = R.residue_field() sage: k Finite Field of size 2 sage: fr_k Ring morphism: From: Finite Field of size 2 To: Valuation ring at Place (x, x*y) sage: to_k Ring morphism: From: Valuation ring at Place (x, x*y) To: Finite Field of size 2 sage: to_k(1/y) 0 sage: to_k(y/(1+y)) 1