exponent valuation

Definition.  A function ν defined in a field K is called an exponent valuation or shortly an exponent of the field, if it satisfies the following conditions:

  1. 1.

    ν(0)=  and ν(α) runs all rational integers when α runs the nonzero elements of K.

  2. 2.


  3. 3.


Note that because of the discrete value set , an exponent valuation belongs to the discrete valuationsPlanetmathPlanetmath, and because of notational causes, to the order valuations.


Example.  If an integral domainMathworldPlanetmath 𝒪 has a divisor theory𝒪*𝔇, then for each prime divisor 𝔭 there is an exponent valuation ν𝔭 of the quotient field K of 𝒪.  It is given by

ν𝔭(α)=:{when α=0,max{k𝔭k(α)} when α0;
ν𝔭(ξ)=:ν𝔭(α)-ν𝔭(β) when ξ=αβ with α,β𝒪*.

Hence, 𝔭ν𝔭(α) exactly divides α.  Apparently, ν𝔭(ξ) does not depend on the quotient form αβ for ξ.  It is not hard to show that ν𝔭 defined above is an exponent of the field K.

Different prime divisors 𝔭 and 𝔮 determine different exponents ν𝔭 and ν𝔮, since the condition 3 of the definition of divisor theory (http://planetmath.org/DivisorTheory) guarantees such an element γ of 𝒪 which in divisible by 𝔭 but not by 𝔮; then  ν𝔭(γ)1,  ν𝔮(γ)=0.

Theorem.  Let  ν1,,νr  be different exponents of a field K.  Then for arbitrary set  n1,,nr  of integers, there exists in K an element ξ such that


The proof of this theorem is found in [1].


  • 1 S. Borewicz & I. Safarevic: Zahlentheorie.  Birkhäuser Verlag. Basel und Stuttgart (1966).
Title exponent valuation
Canonical name ExponentValuation
Date of creation 2013-03-22 17:59:31
Last modified on 2013-03-22 17:59:31
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 12
Author pahio (2872)
Entry type Definition
Classification msc 13F30
Classification msc 13A18
Classification msc 12J20
Classification msc 11R99
Synonym exponent of field
Related topic DiscreteValuation
Related topic OrderValuation
Related topic UltrametricTriangleInequality
Related topic DivisorTheoryAndExponentValuations
Related topic DivisorTheory
Defines exponent of a field
Defines exponent of the field