valuation determined by valuation domain
Theorem.
Every valuation domain determines a Krull valuation of the field of fractions.
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 subgroup of the multiplicative group K*=K∖{0}. So we can form the factor group K*/E, consisting of all cosets aE where a∈K*, and attach to it the additional “coset” 0E getting thus a multiplicative group K/E equipped with zero. If 𝔪=R∖E is the maximal ideal 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:a∈𝔪*}. 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
x↦|x|:=xE |
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 |