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

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

Python

>>> from sage.all import *
>>> K = FunctionField(GF(Integer(2)), names=('x',)); (x,) = K._first_ngens(1); _ = K['Y']; (Y,) = _._first_ngens(1)
>>> L = K.extension(Y**Integer(2) + Y + x + Integer(1)/x, names=('y',)); (y,) = L._first_ngens(1)
>>> p = L.places_finite()[Integer(0)]
>>> p
Place (x, x*y)
>>> R = p.valuation_ring()
>>> R
Valuation ring at Place (x, x*y)
>>> R.place() == p
True

Sage Live

K.<x> = FunctionField(GF(2)); _.<Y> = K[]
L.<y> = K.extension(Y^2 + Y + x + 1/x)
p = L.places_finite()[0]
p
R = p.valuation_ring()
R
R.place() == p

Thus any nonzero element or its inverse of the function field lies in the valuation ring, as shown in the following example:

Sage

sage: f = y/(1+y)
sage: f in R
True
sage: f not in R
False
sage: f.valuation(p)
0

Python

>>> from sage.all import *
>>> f = y/(Integer(1)+y)
>>> f in R
True
>>> f not in R
False
>>> f.valuation(p)
0

Sage Live

f = y/(1+y)
f in R
f not in R
f.valuation(p)

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

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

Python

>>> from sage.all import *
>>> k,phi,psi = R.residue_field()
>>> k
Finite Field of size 2
>>> phi
Ring morphism:
  From: Finite Field of size 2
  To:   Valuation ring at Place (x, x*y)
>>> psi
Ring morphism:
  From: Valuation ring at Place (x, x*y)
  To:   Finite Field of size 2
>>> psi(f) in k
True

Sage Live

k,phi,psi = R.residue_field()
k
phi
psi
psi(f) in k

AUTHORS:

• Kwankyu Lee (2017-04-30): initial version

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

Bases: UniqueRepresentation, Parent

Base class for valuation rings of function fields.

INPUT:

• field – function field

• place – place of the function field

EXAMPLES:

Sage

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)

Python

>>> from sage.all import *
>>> K = FunctionField(GF(Integer(2)), names=('x',)); (x,) = K._first_ngens(1); _ = K['Y']; (Y,) = _._first_ngens(1)
>>> L = K.extension(Y**Integer(2) + Y + x + Integer(1)/x, names=('y',)); (y,) = L._first_ngens(1)
>>> p = L.places_finite()[Integer(0)]
>>> p.valuation_ring()
Valuation ring at Place (x, x*y)

Sage Live

K.<x> = FunctionField(GF(2)); _.<Y> = K[]
L.<y> = K.extension(Y^2 + Y + x + 1/x)
p = L.places_finite()[0]
p.valuation_ring()

place()

Return the place associated with the valuation ring.

EXAMPLES:

Sage

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

Python

>>> from sage.all import *
>>> K = FunctionField(GF(Integer(2)), names=('x',)); (x,) = K._first_ngens(1); _ = K['Y']; (Y,) = _._first_ngens(1)
>>> L = K.extension(Y**Integer(2) + Y + x + Integer(1)/x, names=('y',)); (y,) = L._first_ngens(1)
>>> p = L.places_finite()[Integer(0)]
>>> R = p.valuation_ring()
>>> p == R.place()
True

Sage Live

K.<x> = FunctionField(GF(2)); _.<Y> = K[]
L.<y> = K.extension(Y^2 + Y + x + 1/x)
p = L.places_finite()[0]
R = p.valuation_ring()
p == R.place()

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

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

Python

>>> from sage.all import *
>>> K = FunctionField(GF(Integer(2)), names=('x',)); (x,) = K._first_ngens(1); _ = K['Y']; (Y,) = _._first_ngens(1)
>>> L = K.extension(Y**Integer(2) + Y + x + Integer(1)/x, names=('y',)); (y,) = L._first_ngens(1)
>>> p = L.places_finite()[Integer(0)]
>>> R = p.valuation_ring()
>>> k, fr_k, to_k = R.residue_field()
>>> k
Finite Field of size 2
>>> fr_k
Ring morphism:
  From: Finite Field of size 2
  To:   Valuation ring at Place (x, x*y)
>>> to_k
Ring morphism:
  From: Valuation ring at Place (x, x*y)
  To:   Finite Field of size 2
>>> to_k(Integer(1)/y)
0
>>> to_k(y/(Integer(1)+y))
1

Sage Live

K.<x> = FunctionField(GF(2)); _.<Y> = K[]
L.<y> = K.extension(Y^2 + Y + x + 1/x)
p = L.places_finite()[0]
R = p.valuation_ring()
k, fr_k, to_k = R.residue_field()
k
fr_k
to_k
to_k(1/y)
to_k(y/(1+y))

Make live