equivalent valuations

Let K be a field.  The equivalence of valuations ||1 and ||2 of K may be defined so that

  1. 1.

    ||1 is not the trivial valuation;

  2. 2.

    if   |a|1<1 then |a|2<1  aK.

It it easy to see that these conditions imply for both valuationsMathworldPlanetmath (use 1a).  Also, we have always


so both valuations have a common valuation ringMathworldPlanetmathPlanetmath in the case they are non-archimedean.  (The of the more general Krull valuations is defined to that they have common valuation rings.)  Further, both valuations determine a common metric on K.


Two valuations (of rank (http://planetmath.org/KrullValuation) one)  ||1  and  ||2  of K are iff one of them is a positive power of the other,

|a|1=|a|2c  aK,

where c is a positive .

Title equivalent valuations
Canonical name EquivalentValuations
Date of creation 2013-03-22 14:25:27
Last modified on 2013-03-22 14:25:27
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 18
Author pahio (2872)
Entry type Definition
Classification msc 13A18
Related topic DiscreteValuation
Related topic IndependenceOfTheValuations
Defines equivalence of valuations