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 factors
.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 factors
.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 theresidue_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 theresidue_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