PlanetMath: Krull valuation

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Krull valuation

(Definition)

Definition. The mapping $\vert.\vert\!:\, K\to G$ , where is a field and an ordered group equipped with zero, is a Krull valuation of , if it has the properties

$\vert x\vert = 0 \,\,\Leftrightarrow\,\, x = 0$ ;
$\vert xy\vert = \vert x\vert\cdot\vert y\vert$ ;
$\vert x+y\vert \leqq \max\{\vert x\vert,\,\vert y\vert\}$ .

Thus the Krull valuation is more general than the usual valuation, which is also characterized as valuation of rank 1 and which has real values. The image $\vert K\!\smallsetminus\!\{0\}\vert$ is called the value group of the Krull valuation; it is abelian. In general, the rank of Krull valuation means the rank of the value group.

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

Some values

$\vert 1\vert = 1$ because the Krull valuation is a group homomorphism from the multiplicative group of to the ordered group.
$\vert-1\vert = 1$ because $1 = \vert(-1)^2\vert = \vert-1\vert^2$ and 1 is the only element of the ordered group being its own inverse ( $S\cap S^{-1} = \varnothing$ ).
$\vert-x\vert = \vert(-1)x\vert = \vert-1\vert\cdot\vert x\vert = \vert x\vert$

Bibliography

1: EMIL ARTIN: Theory of Algebraic Numbers. Lecture notes. Mathematisches Institut, Göttingen (1959).

"Krull valuation" is owned by pahio.

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Also defines:

value group, rank of Krull valuation, rank of valuation

This object's parent.

Attachments:: ultrametric triangle inequality (Theorem) by pahio
valuation determined by valuation domain (Theorem) by pahio
Krull valuation domain (Theorem) by pahio
extension of Krull valuation (Theorem) by pahio
order valuation (Definition) by pahio

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Cross-references: inverse, ordered group, multiplicative group, group homomorphism, abelian, image, real, properties, ordered group equipped with zero, field, mapping
There are 10 references to this entry.

This is version 14 of Krull valuation, born on 2004-12-27, modified 2007-04-05.
Object id is 6596, canonical name is KrullValuation.
Accessed 3816 times total.

Classification:

AMS MSC:	11R99 (Number theory :: Algebraic number theory: global fields :: Miscellaneous)
	12J20 (Field theory and polynomials :: Topological fields :: General valuation theory)
	13A18 (Commutative rings and algebras :: General commutative ring theory :: Valuations and their generalizations)
	13F30 (Commutative rings and algebras :: Arithmetic rings and other special rings :: Valuation rings)

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