Valuation

Remarks

Example: for a prime $p$, the p-adic valuation $v_p$ on $\Q$ with values in $\Z$ is a valuation. If $x = p^m a/b$ and $y = p^n c/d$ with $a, b, c, d$ prime to $p$ and $m \le n$, then $xy = p^{m+n} ac/(bd)$ gives clause 2, and $x + y = p^m (ad + p^{n-m} cb)/(bd)$ gives $v_p(x + y) \ge m$, which is clause 3. Example: on the field $k(t)$ of rational functions over a field $k$, the map $f/g \mapsto \deg g - \deg f$ is a valuation with values in $\Z$. Non-example: the map $v \colon \Q \to \Z \cup \set{\infty}$ with $v(0) = \infty$ and $v(x) = 1$ for $x \ne 0$ fails clause 2, since $v(1 \cdot 1) = 1 \ne 2 = v(1) + v(1)$. Every field has the trivial valuation, with $v(x) = 0$ for all $x \ne 0$. Clauses 1 and 2 give $v(1) = 0$, $v(-x) = v(x)$ and $v(x^{-1}) = -v(x)$ for $x \ne 0$. The set $\set{x \in K : v(x) \ge 0}$ is a valuation ring of $K$. Some texts write valuations multiplicatively, with the reverse order; for $\Gamma \subseteq \R$ the map $x \mapsto c^{v(x)}$ with $0 < c < 1$ is a non-Archimedean absolute value.