Valuations which are scaled versions of another valuation#

EXAMPLES:

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

AUTHORS:

  • Julian RĂ¼th (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