Absolute value of a field (reviewed)

An absolute value of a field $k$ is a function $|\ |:k\to \R_{\ge 0}$ that satisfies:

• $|x|=0$ if and only if $x=0$;
• $|xy| = |x||y|$;
• $|x+y| \le |x|+|y|$.

Absolute values that satisfy the stronger condition $|x+y|\le \max(|x|,|y|)$ are nonarchimedean, while those that do not are archimedean; the latter arise only in fields of characteristic zero. The trivial absolute value assigns 1 to every nonzero element of $k$; it is a nonarchimedean absolute value.

Absolute values $|\ |_1$ and $|\ |_2$ are equivalent if there exists a positive real number $c$ such that $|x|_1 = |x|_2^c$ for all $x\in k$; this defines an equivalence relation on the set of absolute values of $k$.