Krull valuation

Definition.  The mapping  |.|:KG,  where K is a field and G an ordered group equipped with zero, is a Krull valuation of K, if it has the properties

  1. 1.


  2. 2.


  3. 3.


Thus the Krull valuation is more general than the usual valuation (, which is also characterized as and which has real values.  The image  |K{0}|  is called the value group of the Krull valuation; it is abelianMathworldPlanetmath.  In general, the rank of Krull valuation the rank ( of the value group.

We may say that a Krull valuation is non-archimedean (

Some values

  • |1|=1   because the Krull valuation is a group homomorphismMathworldPlanetmath from the multiplicative groupMathworldPlanetmath of K to the ordered group.

  • |-1|=1   because  1=|(-1)2|=|-1|2   and 1 is the only element of the ordered group being its own inverseMathworldPlanetmathPlanetmathPlanetmathPlanetmath (SS-1=).

  • |-x|=|(-1)x|=|-1||x|=|x|


  • 1 Emil Artin: Theory of Algebraic NumbersMathworldPlanetmath.  Lecture notes.  Mathematisches Institut, Göttingen (1959).
  • 2 P. Jaffard: Les systèmes d’idéaux.  Dunod, Paris (1960).
Title Krull valuation
Canonical name KrullValuation
Date of creation 2013-03-22 14:54:39
Last modified on 2013-03-22 14:54:39
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 19
Author pahio (2872)
Entry type Definition
Classification msc 13F30
Classification msc 13A18
Classification msc 12J20
Classification msc 11R99
Related topic OrderedGroup
Related topic TrivialValuation
Related topic IsolatedSubgroup
Related topic ValueGroupOfCompletion
Related topic PlaceOfField
Related topic OrderValuation
Related topic AlternativeDefinitionOfValuation2
Related topic UniquenessOfDivisionAlgorithmInEuclideanDomain
Defines value group
Defines rank of Krull valuation
Defines rank of valuation