Valuations which are scaled versions of another valuation

EXAMPLES:

sage: 3*ZZ.valuation(3)
3 * 3-adic valuation

AUTHORS:

• Julian Rüth (2016-11-10): initial version

class sage.rings.valuation.scaled_valuation.ScaledValuationFactory

Bases: UniqueFactory

Return a valuation which scales the valuation base by the factor s.

EXAMPLES:

sage: 3*ZZ.valuation(2) # indirect doctest
3 * 2-adic valuation

create_key(base, s)

Create a key which uniquely identifies a valuation.

create_object(version, key)

Create a valuation from key.

class sage.rings.valuation.scaled_valuation.ScaledValuation_generic(parent, base_valuation, s)

Bases: DiscreteValuation

A valuation which scales another base_valuation by a finite positive factor s.

EXAMPLES:

sage: v = 3*ZZ.valuation(3); v
3 * 3-adic valuation

extensions(ring)

Return the extensions of this valuation to ring.

EXAMPLES:

sage: v = 3*ZZ.valuation(5)
sage: v.extensions(GaussianIntegers().fraction_field())                     # needs sage.rings.number_field
[3 * [ 5-adic valuation, v(x + 2) = 1 ]-adic valuation,
 3 * [ 5-adic valuation, v(x + 3) = 1 ]-adic valuation]

lift(F)

Lift F from the residue_field() of this valuation into its domain.

EXAMPLES:

sage: v = 3*ZZ.valuation(2)
sage: v.lift(1)
1

reduce(f)

Return the reduction of f in the residue_field() of this valuation.

EXAMPLES:

sage: v = 3*ZZ.valuation(2)
sage: v.reduce(1)
1

residue_ring()

Return the residue field of this valuation.

EXAMPLES:

sage: v = 3*ZZ.valuation(2)
sage: v.residue_ring()
Finite Field of size 2

restriction(ring)

Return the restriction of this valuation to ring.

EXAMPLES:

sage: v = 3*QQ.valuation(5)
sage: v.restriction(ZZ)
3 * 5-adic valuation

uniformizer()

Return a uniformizing element of this valuation.

EXAMPLES:

sage: v = 3*ZZ.valuation(2)
sage: v.uniformizer()
2

value_semigroup()

Return the value semigroup of this valuation.

EXAMPLES:

sage: v2 = QQ.valuation(2)
sage: (2*v2).value_semigroup()
Additive Abelian Semigroup generated by -2, 2