Trivial valuations¶
AUTHORS:
- Julian Rüth (2016-10-14): initial version 
EXAMPLES:
sage: v = valuations.TrivialValuation(QQ); v
Trivial valuation on Rational Field
sage: v(1)
0
>>> from sage.all import *
>>> v = valuations.TrivialValuation(QQ); v
Trivial valuation on Rational Field
>>> v(Integer(1))
0
- class sage.rings.valuation.trivial_valuation.TrivialDiscretePseudoValuation(parent)[source]¶
- Bases: - TrivialDiscretePseudoValuation_base,- InfiniteDiscretePseudoValuation- The trivial pseudo-valuation that is \(\infty\) everywhere. - EXAMPLES: - sage: v = valuations.TrivialPseudoValuation(QQ); v Trivial pseudo-valuation on Rational Field - >>> from sage.all import * >>> v = valuations.TrivialPseudoValuation(QQ); v Trivial pseudo-valuation on Rational Field - lift(X)[source]¶
- Return a lift of - Xto the domain of this valuation.- EXAMPLES: - sage: v = valuations.TrivialPseudoValuation(QQ) sage: v.lift(v.residue_ring().zero()) 0 - >>> from sage.all import * >>> v = valuations.TrivialPseudoValuation(QQ) >>> v.lift(v.residue_ring().zero()) 0 
 - reduce(x)[source]¶
- Reduce - xmodulo the positive elements of this valuation.- EXAMPLES: - sage: v = valuations.TrivialPseudoValuation(QQ) sage: v.reduce(1) 0 - >>> from sage.all import * >>> v = valuations.TrivialPseudoValuation(QQ) >>> v.reduce(Integer(1)) 0 
 - residue_ring()[source]¶
- Return the residue ring of this valuation. - EXAMPLES: - sage: valuations.TrivialPseudoValuation(QQ).residue_ring() Quotient of Rational Field by the ideal (1) - >>> from sage.all import * >>> valuations.TrivialPseudoValuation(QQ).residue_ring() Quotient of Rational Field by the ideal (1) 
 - value_group()[source]¶
- Return the value group of this valuation. - EXAMPLES: - A trivial discrete pseudo-valuation has no value group: - sage: v = valuations.TrivialPseudoValuation(QQ) sage: v.value_group() Traceback (most recent call last): ... ValueError: The trivial pseudo-valuation that is infinity everywhere does not have a value group. - >>> from sage.all import * >>> v = valuations.TrivialPseudoValuation(QQ) >>> v.value_group() Traceback (most recent call last): ... ValueError: The trivial pseudo-valuation that is infinity everywhere does not have a value group. 
 
- class sage.rings.valuation.trivial_valuation.TrivialDiscretePseudoValuation_base(parent)[source]¶
- Bases: - DiscretePseudoValuation- Base class for code shared by trivial valuations. - EXAMPLES: - sage: v = valuations.TrivialPseudoValuation(ZZ); v Trivial pseudo-valuation on Integer Ring - >>> from sage.all import * >>> v = valuations.TrivialPseudoValuation(ZZ); v Trivial pseudo-valuation on Integer Ring - is_negative_pseudo_valuation()[source]¶
- Return whether this valuation attains the value \(-\infty\). - EXAMPLES: - sage: v = valuations.TrivialPseudoValuation(QQ) sage: v.is_negative_pseudo_valuation() False - >>> from sage.all import * >>> v = valuations.TrivialPseudoValuation(QQ) >>> v.is_negative_pseudo_valuation() False 
 - is_trivial()[source]¶
- Return whether this valuation is trivial. - EXAMPLES: - sage: v = valuations.TrivialPseudoValuation(QQ) sage: v.is_trivial() True - >>> from sage.all import * >>> v = valuations.TrivialPseudoValuation(QQ) >>> v.is_trivial() True 
 - uniformizer()[source]¶
- Return a uniformizing element for this valuation. - EXAMPLES: - sage: v = valuations.TrivialPseudoValuation(ZZ) sage: v.uniformizer() Traceback (most recent call last): ... ValueError: Trivial valuations do not define a uniformizing element - >>> from sage.all import * >>> v = valuations.TrivialPseudoValuation(ZZ) >>> v.uniformizer() Traceback (most recent call last): ... ValueError: Trivial valuations do not define a uniformizing element 
 
- class sage.rings.valuation.trivial_valuation.TrivialDiscreteValuation(parent)[source]¶
- Bases: - TrivialDiscretePseudoValuation_base,- DiscreteValuation- The trivial valuation that is zero on nonzero elements. - EXAMPLES: - sage: v = valuations.TrivialValuation(QQ); v Trivial valuation on Rational Field - >>> from sage.all import * >>> v = valuations.TrivialValuation(QQ); v Trivial valuation on Rational Field - extensions(ring)[source]¶
- Return the unique extension of this valuation to - ring.- EXAMPLES: - sage: v = valuations.TrivialValuation(ZZ) sage: v.extensions(QQ) [Trivial valuation on Rational Field] - >>> from sage.all import * >>> v = valuations.TrivialValuation(ZZ) >>> v.extensions(QQ) [Trivial valuation on Rational Field] 
 - lift(X)[source]¶
- Return a lift of - Xto the domain of this valuation.- EXAMPLES: - sage: v = valuations.TrivialValuation(QQ) sage: v.lift(v.residue_ring().zero()) 0 - >>> from sage.all import * >>> v = valuations.TrivialValuation(QQ) >>> v.lift(v.residue_ring().zero()) 0 
 - reduce(x)[source]¶
- Reduce - xmodulo the positive elements of this valuation.- EXAMPLES: - sage: v = valuations.TrivialValuation(QQ) sage: v.reduce(1) 1 - >>> from sage.all import * >>> v = valuations.TrivialValuation(QQ) >>> v.reduce(Integer(1)) 1 
 - residue_ring()[source]¶
- Return the residue ring of this valuation. - EXAMPLES: - sage: valuations.TrivialValuation(QQ).residue_ring() Rational Field - >>> from sage.all import * >>> valuations.TrivialValuation(QQ).residue_ring() Rational Field 
 - value_group()[source]¶
- Return the value group of this valuation. - EXAMPLES: - A trivial discrete valuation has a trivial value group: - sage: v = valuations.TrivialValuation(QQ) sage: v.value_group() Trivial Additive Abelian Group - >>> from sage.all import * >>> v = valuations.TrivialValuation(QQ) >>> v.value_group() Trivial Additive Abelian Group 
 
- class sage.rings.valuation.trivial_valuation.TrivialValuationFactory(clazz, parent, *args, **kwargs)[source]¶
- Bases: - UniqueFactory- Create a trivial valuation on - domain.- EXAMPLES: - sage: v = valuations.TrivialValuation(QQ); v Trivial valuation on Rational Field sage: v(1) 0 - >>> from sage.all import * >>> v = valuations.TrivialValuation(QQ); v Trivial valuation on Rational Field >>> v(Integer(1)) 0 - create_key(domain)[source]¶
- Create a key that identifies this valuation. - EXAMPLES: - sage: valuations.TrivialValuation(QQ) is valuations.TrivialValuation(QQ) # indirect doctest True - >>> from sage.all import * >>> valuations.TrivialValuation(QQ) is valuations.TrivialValuation(QQ) # indirect doctest True 
 - create_object(version, key, **extra_args)[source]¶
- Create a trivial valuation from - key.- EXAMPLES: - sage: valuations.TrivialValuation(QQ) # indirect doctest Trivial valuation on Rational Field - >>> from sage.all import * >>> valuations.TrivialValuation(QQ) # indirect doctest Trivial valuation on Rational Field