PlanetMath: valuation determined by valuation domain

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valuation determined by valuation domain

(Theorem)

Theorem 1 Every valuation domain determines a Krull valuation of the field of fractions.

Proof. Let be a valuation domain, its field of fractions and the group of units of . Then is a normal subgroup of the multiplicative group $K^* = K\!\smallsetminus\!\{0\}$ . So we can form the factor group , consisting of all cosets where $a\in K^*$ , and attach to it the additional “coset” getting thus a multiplicative group equipped with zero. If $\mathfrak{m} = R\!\smallsetminus\!E$ is the maximal ideal of (any valuation domain has a unique maximal ideal -- cf. valuation domain is local), then we denote $\mathfrak{m}^* = \mathfrak{m}\!\smallsetminus\!\{0\}$ and $S = \mathfrak{m}^*/E = \{aE:\,\,a\in \mathfrak{m}^*\}$ . Then the subsemigroup of makes an ordered group equipped with zero. It is not hard to check that the mapping

$\displaystyle x\mapsto \vert x\vert := xE$

from

is a Krull valuation of the field

"valuation determined by valuation domain" is owned by pahio.

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See Also: valuation domain is local, Krull valuation domain, place of field

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Cross-references: field, mapping, ordered group equipped with zero, subsemigroup, valuation domain is local, maximal ideal, cosets, factor group, multiplicative group, normal subgroup, group of units, field of fractions, Krull valuation, valuation domain

This is version 7 of valuation determined by valuation domain, born on 2004-12-28, modified 2006-12-26.
Object id is 6602, canonical name is ValuationDeterminedByValuationDomain.
Accessed 959 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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