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. In a global function field, the residue_field() method returns a finite field isomorphic to the residue field:

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 field
• place – 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

class sage.rings.function_field.valuation_ring.FunctionFieldValuationRing_global(field, place, category=None)

Bases: sage.rings.function_field.valuation_ring.FunctionFieldValuationRing

Valuation rings of global function fields.

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 residue field

OUTPUT:

• a finite field isomorphic to the residue field
• a ring homomorphism from the valuation ring to the finite field
• a ring homomorphism from the finite 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