integral element
An element $a$ of a field $K$ is an integral element^{} of the field $K$, iff
$$a\beta \x89\u20ac1$$ 
for every nonarchimedean valuationβ $\beta \x8b\x85$β of this field.
The set $\mathrm{\pi \x9d\x92\u037a}$ of all integral elements of $K$ is a subring (in fact, an integral domain^{}) of $K$, because it is the intersection of all valuation rings^{} in $K$.
Examples

1.
$K=\mathrm{\beta \x84\x9a}$.β The only nonarchimedean valuations of $\mathrm{\beta \x84\x9a}$ are the $p$adic valuations^{}β $\beta \x8b\x85{}_{p}$β (where $p$ is a rational prime) and the trivial valuation (all values are 1 except the value of 0).β The valuation ring ${\mathrm{\pi \x9d\x92\u037a}}_{p}$ ofβ $\beta \x8b\x85{}_{p}$β consists of all socalled pintegral rational numbers whose denominators are not divisible by $p$.β The valuation ring of the trivial valuation is, generally, the whole field.β Thus, $\mathrm{\pi \x9d\x92\u037a}$ is, by definition, the intersection of the ${\mathrm{\pi \x9d\x92\u037a}}_{p}$βs for all $p$;β this is the set of rationals whose denominators are not divisible by any prime, which is exactly the set $\mathrm{\beta \x84\u20ac}$ of ordinary integers.

2.
If $K$ is a finite field^{}, it has only the trivial valuation.β In fact, if $\beta \x8b\x85$ is a valuation and $a$ any nonzero element of $K$, then there is a positive integer $m$ such thatβ ${a}^{m}=1$,β and we haveβ ${a}^{m}={a}^{m}=1=1$,β and thereforeβ $a=1$.β Thus, $\beta \x8b\x85$ is trivial andβ $\mathrm{\pi \x9d\x92\u037a}=K$.β This means that all elements of the field are integral elements.

3.
If $K$ is the field ${\mathrm{\beta \x84\x9a}}_{p}$ of the $p$adic numbers (http://planetmath.org/NonIsomorphicCompletionsOfMathbbQ), it has only one nontrivial valuation, the $p$adic valuation, and now the ring $\mathrm{\pi \x9d\x92\u037a}$ is its valuation ring, which is the ring of $p$adic integers (http://planetmath.org/PAdicIntegers);β this is visualized in the 2adic (dyadic) case in the article β$p$adic canonical formβ.
Title  integral element 

Canonical name  IntegralElement 
Date of creation  20130322 14:15:56 
Last modified on  20130322 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 