Valuations which are scaled versions of another valuation¶
EXAMPLES:
sage: 3*ZZ.valuation(3)
3 * 3-adic valuation
>>> from sage.all import *
>>> Integer(3)*ZZ.valuation(Integer(3))
3 * 3-adic valuation
AUTHORS:
- Julian Rüth (2016-11-10): initial version 
- class sage.rings.valuation.scaled_valuation.ScaledValuationFactory[source]¶
- Bases: - UniqueFactory- Return a valuation which scales the valuation - baseby the factor- s.- EXAMPLES: - sage: 3*ZZ.valuation(2) # indirect doctest 3 * 2-adic valuation - >>> from sage.all import * >>> Integer(3)*ZZ.valuation(Integer(2)) # indirect doctest 3 * 2-adic valuation 
- class sage.rings.valuation.scaled_valuation.ScaledValuation_generic(parent, base_valuation, s)[source]¶
- Bases: - DiscreteValuation- A valuation which scales another - base_valuationby a finite positive factor- s.- EXAMPLES: - sage: v = 3*ZZ.valuation(3); v 3 * 3-adic valuation - >>> from sage.all import * >>> v = Integer(3)*ZZ.valuation(Integer(3)); v 3 * 3-adic valuation - extensions(ring)[source]¶
- 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] - >>> from sage.all import * >>> v = Integer(3)*ZZ.valuation(Integer(5)) >>> 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)[source]¶
- Lift - Ffrom the- residue_field()of this valuation into its domain.- EXAMPLES: - sage: v = 3*ZZ.valuation(2) sage: v.lift(1) 1 - >>> from sage.all import * >>> v = Integer(3)*ZZ.valuation(Integer(2)) >>> v.lift(Integer(1)) 1 
 - reduce(f)[source]¶
- Return the reduction of - fin the- residue_field()of this valuation.- EXAMPLES: - sage: v = 3*ZZ.valuation(2) sage: v.reduce(1) 1 - >>> from sage.all import * >>> v = Integer(3)*ZZ.valuation(Integer(2)) >>> v.reduce(Integer(1)) 1 
 - residue_ring()[source]¶
- Return the residue field of this valuation. - EXAMPLES: - sage: v = 3*ZZ.valuation(2) sage: v.residue_ring() Finite Field of size 2 - >>> from sage.all import * >>> v = Integer(3)*ZZ.valuation(Integer(2)) >>> v.residue_ring() Finite Field of size 2 
 - restriction(ring)[source]¶
- Return the restriction of this valuation to - ring.- EXAMPLES: - sage: v = 3*QQ.valuation(5) sage: v.restriction(ZZ) 3 * 5-adic valuation - >>> from sage.all import * >>> v = Integer(3)*QQ.valuation(Integer(5)) >>> v.restriction(ZZ) 3 * 5-adic valuation 
 - uniformizer()[source]¶
- Return a uniformizing element of this valuation. - EXAMPLES: - sage: v = 3*ZZ.valuation(2) sage: v.uniformizer() 2 - >>> from sage.all import * >>> v = Integer(3)*ZZ.valuation(Integer(2)) >>> v.uniformizer() 2 
 - value_semigroup()[source]¶
- 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 - >>> from sage.all import * >>> v2 = QQ.valuation(Integer(2)) >>> (Integer(2)*v2).value_semigroup() Additive Abelian Semigroup generated by -2, 2