order valuation

Given a Krull valuation |.| of a field K as a mapping of K to an ordered group G (with operation “”) equipped with 0, one may use for the an alternative notation “ord”:

The “<” of G is reversed and the operation of G is denoted by “+”.  The element 0 of G is denoted as , thus is greater than any other element of G.  When we still call the valuationMathworldPlanetmath the order of K and instead of |x| write  ordx, the valuation postulates read as follows.

  1. 1.

    ordx=   iff   x=0;

  2. 2.


  3. 3.


We must emphasize that the order valuation is nothing else than a Krull valuation.  The name order comes from complex analysis, where the “places” zero (http://planetmath.org/ZeroOfAFunction) and pole (http://planetmath.org/Pole) of a meromorphic function with their orders have a fully analogous meaning as have the corresponding concepts place (http://planetmath.org/PlaceOfField) and order valuation in the valuation theory.  Thus also a place φ of a field is called a zero of an element a of the field, if  φ(a)=0,  and a pole of an element b of the field, if  φ(b)=;  then e.g. the equationφ(a)=0  implies always that  orda>0.

Example.  Let p be a given positive prime numberMathworldPlanetmath.  Any non-zero rational number x can be uniquely expressed in the form


in which n is an integer and the rational number u is by p indivisible, i.e. when reduced to lowest terms, p divides neither its numerator nor its denominator.  If we define


then the function ordp, defined in , clearly satisfies the above postulates of the order valuation.

In [2], an order valuation having only integer values is called the exponentPlanetmathPlanetmath of the field (der Exponent des Körpers); this name apparently motivated by the exponent n of p.  Such an order valuation is a special case of the discrete valuationPlanetmathPlanetmath.  Note, that an arbitrary order valuation need not be a discrete valuation, since the values need not be real numbers.


  • 1 E. Artin: Theory of Algebraic NumbersMathworldPlanetmath.  Lecture notes.  Mathematisches Institut, Göttingen (1959).
  • 2 S. Borewicz & I. Safarevic: Zahlentheorie.  Birkhäuser Verlag. Basel und Stuttgart (1966).
Title order valuation
Canonical name OrderValuation
Date of creation 2013-03-22 16:53:28
Last modified on 2013-03-22 16:53:28
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 19
Author pahio (2872)
Entry type Definition
Classification msc 13F30
Classification msc 13A18
Classification msc 12J20
Classification msc 11R99
Synonym additive valuation
Related topic KrullValuation
Related topic Valuation
Related topic PAdicValuation
Related topic DiscreteValuation
Related topic ZerosAndPolesOfRationalFunction
Related topic AlternativeDefinitionOfValuation2
Related topic StrictDivisibility
Related topic ExponentValuation2
Related topic DivisibilityOfPrimePowerBinomialCoefficients
Defines exponent of fieldPlanetmathPlanetmath
Defines zero
Defines zero of an element
Defines pole
Defines pole of an element