Let $K$ be a field. A valuation or absolute value on $K$ is a function $|\cdot|\colon K \to \R$ satisfying the properties:

1. $|x| \geq 0$ for all $x \in K$ , with equality if and only if $x=0$
2. $|xy| = |x|\cdot |y|$ for all $x,y \in K$
3. $|x+y| \leq |x| + |y|$

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 equivalent if their corresponding metrics induce the same topology on $K$ . An equivalence class $v$ 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 archimedean prime. Otherwise, we say that $v$ is a finite prime, or non-archimedean prime.

In the case where $K$ is a number field, primes as defined above generalize the notion of prime ideals in the following way. Let $\p \subset K$ be a nonzero prime ideal ¹, considered as a fractional ideal. For every nonzero element $x \in K$ , let $r$ be the unique integer such that $x \in \p^r$ but $x \notin \p^{r+1}$ . Define

As for the archimedean valuations, when $K$ is a number field every embedding of $K$ into $\R$ or $\C$ yields a valuation of $K$ by way of the standard absolute value on $\R$ or $\C$ , 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 $\R$ or $\C$ . Such a prime is called real or complex according to whether the valuations comprising it arise from real or complex embeddings.

Footnotes

... ideal¹

By ``prime ideal'' we mean ``prime fractional ideal of $K$ '' or equivalently ``prime ideal of the ring of integers of $K$ ''. We do not mean literally a prime ideal of the ring $K$ , which would be the zero ideal.