Krull valuation
Definition. The mapping |.|:K→G, 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.
|x|=0⇔x=0;
-
2.
|xy|=|x|⋅|y|;
-
3.
|x+y|≦.
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
-
•
because the Krull valuation is a group homomorphism
from the multiplicative group
of to the ordered group.
-
•
because and 1 is the only element of the ordered group being its own inverse
().
-
•
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 |