Krull valuation
Definition. The mapping , where is a field and an ordered group equipped with zero, is a Krull valuation of , if it has the properties
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1.
;
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2.
;
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3.
.
Thus the Krull valuation is more general than the usual valuation (http://planetmath.org/Valuation), which is also characterized as and which has real values. The image is called the value group of the Krull valuation; it is abelian. In general, the rank of Krull valuation the rank (http://planetmath.org/IsolatedSubgroup) of the value group.
We may say that a Krull valuation is non-archimedean (http://planetmath.org/Valuation).
Some values
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•
because the Krull valuation is a group homomorphism from the multiplicative group of to the ordered group.
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•
because and 1 is the only element of the ordered group being its own inverse ().
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•
References
- 1 Emil Artin: Theory of Algebraic Numbers. 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 |