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
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class
sage.rings.valuation.scaled_valuation.ScaledValuationFactory¶ Bases:
sage.structure.factory.UniqueFactoryReturn a valuation which scales the valuation
baseby the factors.EXAMPLES:
sage: 3*ZZ.valuation(2) # indirect doctest 3 * 2-adic valuation
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create_key(base, s)¶ Create a key which uniquely identifies a valuation.
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create_object(version, key)¶ Create a valuation from
key.
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class
sage.rings.valuation.scaled_valuation.ScaledValuation_generic(parent, base_valuation, s)¶ Bases:
sage.rings.valuation.valuation.DiscreteValuationA valuation which scales another
base_valuationby a finite positive factors.EXAMPLES:
sage: v = 3*ZZ.valuation(3); v 3 * 3-adic valuation
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extensions(ring)¶ Return the extensions of this valuation to
ring.EXAMPLES:
sage: v = 3*ZZ.valuation(5) sage: v.extensions(GaussianIntegers().fraction_field()) [3 * [ 5-adic valuation, v(x + 2) = 1 ]-adic valuation, 3 * [ 5-adic valuation, v(x + 3) = 1 ]-adic valuation]
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lift(F)¶ Lift
Ffrom theresidue_field()of this valuation into its domain.EXAMPLES:
sage: v = 3*ZZ.valuation(2) sage: v.lift(1) 1
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reduce(f)¶ Return the reduction of
fin theresidue_field()of this valuation.EXAMPLES:
sage: v = 3*ZZ.valuation(2) sage: v.reduce(1) 1
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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
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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
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uniformizer()¶ Return a uniformizing element of this valuation.
EXAMPLES:
sage: v = 3*ZZ.valuation(2) sage: v.uniformizer() 2
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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
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