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

1. 1.

   $|x|\geq 0$ for all $x\in K$, with equality if and only if $x=0$

2. 2.

   $|xy|=|x|\cdot|y|$ for all $x,y\in K$

3. 3.

   $|x+y|\leq|x|+|y|$

If a valuation satisfies $|x+y|\leq\max(|x|,|y|)$, 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 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 $\mathfrak{p}\subset K$ be a nonzero prime ideal¹¹By “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., considered as a fractional ideal. For every nonzero element $x\in K$, let $r$ be the unique integer such that $x\in\mathfrak{p}^{r}$ but $x\notin\mathfrak{p}^{{r+1}}$. Define

where $N(\mathfrak{p})$ denotes the absolute norm of $\mathfrak{p}$. Then $|\cdot|_{\mathfrak{p}}$ is a non–archimedean valuation on $K$, and furthermore every non-archimedean valuation on $K$ is equivalent to $|\cdot|_{\mathfrak{p}}$ for some prime ideal $\mathfrak{p}$. 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 embedding of $K$ into $\mathbb{R}$ or $\mathbb{C}$ yields a valuation of $K$ by way of the standard absolute value on $\mathbb{R}$ or $\mathbb{C}$, 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 $\mathbb{R}$ or $\mathbb{C}$. Such a prime is called real or complex according to whether the valuations comprising it arise from real or complex embeddings.