Krull valuation domain


Any Krull valuation||  of a field K determines a unique valuation domainR={aK:|x|1}, whose field of fractionMathworldPlanetmath is K.

Proof.  We first see that  1R  since  |1|=1.  Let then  a,b  be any two elements of R.  The non-archimedean triangle inequality shows that  |a-b|max{|a|,|b|}1,  i.e. that the differencea-b  belongs to R.  Using the multiplication rule ( 4 of inequalities we obtain


which shows that also the product ab is element of R.  Thus, R is a subring of the field K, and so an integral domainMathworldPlanetmath.  Let now c be an arbitrary element of K not belonging to R.  This implies that  1<|c|,  whence  |c-1|=|c|-1<1 (see the inverse rule ( 5).  Consequently, the inverse c-1 belongs to R, and we conclude that R is a valuation domain.   The  a=a1  and  c=1c-1  make evident that K is the field of fractions of R.

Title Krull valuation domain
Canonical name KrullValuationDomain
Date of creation 2013-03-22 14:55:01
Last modified on 2013-03-22 14:55:01
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 8
Author pahio (2872)
Entry type Theorem
Classification msc 13F30
Classification msc 13A18
Classification msc 12J20
Classification msc 11R99
Related topic ValuationDeterminedByValuationDomain