valuation determined by valuation domain


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


from K to  K/E  is a Krull valuation of the field K.

Title valuation determined by valuation domain
Canonical name ValuationDeterminedByValuationDomain
Date of creation 2013-03-22 14:54:58
Last modified on 2013-03-22 14:54:58
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 10
Author pahio (2872)
Entry type Theorem
Classification msc 13F30
Classification msc 13A18
Classification msc 12J20
Classification msc 11R99
Related topic ValuationDomainIsLocal
Related topic KrullValuationDomain
Related topic PlaceOfField