integral element

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

$|a|\leq 1$

for every non-archimedean valuation $|\cdot|$ of this field.

The set $\mathcal{O}$ 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.

$K=\mathbb{Q}$ . The only non-archimedean valuations of $\mathbb{Q}$ are the $p$ -adic valuations $|\cdot|_{p}$ (where $p$ is a rational prime) and the trivial valuation (all values are 1 except the value of 0). The valuation ring $\mathcal{O}_{p}$ of $|\cdot|_{p}$ 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, $\mathcal{O}$ is, by definition, the intersection of the $\mathcal{O}_{p}$ ’s for all $p$ ; this is the set of rationals whose denominators are not divisible by any prime, which is exactly the set $\mathbb{Z}$ of ordinary integers.
2.

If $K$ is a finite field, it has only the trivial valuation. In fact, if $|\cdot|$ is a valuation and $a$ any non-zero element of $K$ , then there is a positive integer $m$ such that $a^{m}=1$ , and we have $|a|^{m}=|a^{m}|=|1|=1$ , and therefore $|a|=1$ . Thus, $|\cdot|$ is trivial and $\mathcal{O}=K$ . This means that all elements of the field are integral elements.
3.

If $K$ is the field $\mathbb{Q}_{p}$ of the $p$ -adic numbers (http://planetmath.org/NonIsomorphicCompletionsOfMathbbQ), it has only one non-trivial valuation, the $p$ -adic valuation, and now the ring $\mathcal{O}$ is its valuation ring, which is the ring of $p$ -adic integers (http://planetmath.org/PAdicIntegers); this is visualized in the 2-adic (dyadic) case in the article “ $p$ -adic canonical form”.