Let K be a field. A valuationMathworldPlanetmathPlanetmath or absolute valueMathworldPlanetmathPlanetmathPlanetmath on K is a function ||:K satisfying the properties:

  1. 1.

    |x|0 for all xK, with equality if and only if x=0

  2. 2.

    |xy|=|x||y| for all x,yK

  3. 3.


If a valuation satisfies |x+y|max(|x|,|y|), then we say that it is a non-archimedean valuation. Otherwise we say that it is an archimedean valuation.

Every valuation on K defines a metric on K, given by d(x,y):=|x-y|. This metric is an ultrametric if and only if the valuation is non-archimedean. Two valuations are equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath if their corresponding metrics induce the same topology on K. An equivalence classMathworldPlanetmath v of valuations on K is called a prime of K. If v consists of archimedean valuations, we say that v is an infinite prime, or archimedeanPlanetmathPlanetmath prime. Otherwise, we say that v is a finite prime, or non-archimedean prime.

In the case where K is a number fieldMathworldPlanetmath, primes as defined above generalize the notion of prime idealsPlanetmathPlanetmathPlanetmath in the following way. Let 𝔭K be a nonzero prime ideal11By “prime ideal” we mean “prime fractional idealMathworldPlanetmathPlanetmath of K” or equivalently “prime ideal of the ring of integersMathworldPlanetmath of K”. We do not mean literally a prime ideal of the ring K, which would be the zero idealPlanetmathPlanetmath., considered as a fractional ideal. For every nonzero element xK, let r be the unique integer such that x𝔭r but x𝔭r+1. Define


where N(𝔭) denotes the absolute norm of 𝔭. Then ||𝔭 is a non–archimedean valuation on K, and furthermore every non-archimedean valuation on K is equivalent to ||𝔭 for some prime ideal 𝔭. Hence, the prime ideals of K correspond bijectively with the finite primes of K, and it is in this sense that the notion of primes as valuations generalizes that of a prime ideal.

As for the archimedean valuations, when K is a number field every embeddingPlanetmathPlanetmath of K into or yields a valuation of K by way of the standard absolute value on or , and one can show that every archimedean valuation of K is equivalent to one arising in this way. Thus the infinite primes of K correspond to embeddings of K into or . Such a prime is called real or complex according to whether the valuations comprising it arise from real or complex embeddings.

Title valuation
Canonical name Valuation
Date of creation 2013-03-22 12:35:07
Last modified on 2013-03-22 12:35:07
Owner djao (24)
Last modified by djao (24)
Numerical id 17
Author djao (24)
Entry type Definition
Classification msc 13F30
Classification msc 13A18
Classification msc 12J20
Classification msc 11R99
Synonym absolute value
Related topic DiscreteValuationRing
Related topic DiscreteValuation
Related topic Ultrametric
Related topic HenselianField
Defines infinite prime
Defines finite prime
Defines archimedean
Defines non-archimedean
Defines real prime
Defines complex prime
Defines prime