integral element

An element a of a field K is an integral elementMathworldPlanetmath of the field K, iff


for every non-archimedean valuation  |β‹…|  of this field.

The set π’ͺ of all integral elements of K is a subring (in fact, an integral domainMathworldPlanetmath) of K, because it is the intersection of all valuation ringsMathworldPlanetmathPlanetmath in K.


  1. 1.

    K=β„š.  The only non-archimedean valuations of β„š are the p-adic valuationsMathworldPlanetmath  |β‹…|p  (where p is a rational prime) and the trivial valuation (all values are 1 except the value of 0).  The valuation ring π’ͺp of  |β‹…|p  consists of all so-called p-integral rational numbers whose denominators are not divisible by p.  The valuation ring of the trivial valuation is, generally, the whole field.  Thus, π’ͺ is, by definition, the intersection of the π’ͺp’s for all p;  this is the set of rationals whose denominators are not divisible by any prime, which is exactly the set β„€ of ordinary integers.

  2. 2.

    If K is a finite fieldMathworldPlanetmath, it has only the trivial valuation.  In fact, if |β‹…| is a valuation and a any non-zero element of K, then there is a positive integer m such that  am=1,  and we have  |a|m=|am|=|1|=1,  and therefore  |a|=1.  Thus, |β‹…| is trivial and  π’ͺ=K.  This means that all elements of the field are integral elements.

  3. 3.

    If K is the field β„šp of the p-adic numbers (, it has only one non-trivial valuation, the p-adic valuation, and now the ring π’ͺ is its valuation ring, which is the ring of p-adic integers (;  this is visualized in the 2-adic (dyadic) case in the article β€œp-adic canonical form”.

Title integral element
Canonical name IntegralElement
Date of creation 2013-03-22 14:15:56
Last modified on 2013-03-22 14:15:56
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 31
Author pahio (2872)
Entry type Definition
Classification msc 12E99
Related topic PAdicCanonicalForm
Related topic PAdicValuation
Related topic KummersCongruence