valuationValuation2002-04-15 18:00:372009-01-07 13:50:23Definitioninfinite primefinite primearchimedeannon-archimedeanreal primecomplex primeprime% this is the default PlanetMath preamble. as your knowledge
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\newcommand{\p}{\mathfrak{p}}Let $K$ be a field. A \emph{valuation} or \emph{absolute value} on $K$ is a function $|\cdot|\colon K \to \R$ satisfying the properties:
\begin{enumerate}
\item $|x| \geq 0$ for all $x \in K$, with equality if and only if $x=0$
\item $|xy| = |x|\cdot |y|$ for all $x,y \in K$
\item $|x+y| \leq |x| + |y|$
\end{enumerate}
If a valuation satisfies $|x+y| \leq \max(|x|, |y|)$, then we say that it is a \emph{non-archimedean valuation}. Otherwise we say that it is an \emph{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 \emph{equivalent} if their corresponding metrics induce the same topology on $K$. An equivalence class $v$ of valuations on $K$ is called a \emph{prime} of $K$. If $v$ consists of archimedean valuations, we say that $v$ is an \emph{infinite prime}, or \emph{archimedean prime}. Otherwise, we say that $v$ is a \emph{finite prime}, or \emph{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\footnote{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.}, 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
$$
|x|_\p :=
\begin{cases}
1/N(\p)^r & x \neq 0, \\
0 & x=0,
\end{cases}
$$
where $N(\p)$ denotes the absolute norm of $\p$. Then $|\cdot|_\p$ is a non--archimedean valuation on $K$, and furthermore every non-archimedean valuation on $K$ is equivalent to $|\cdot|_\p$ for some prime ideal $\p$. 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 $\R$ 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.