Definition:Valuation

Definition
Let $\left({k, +, \cdot}\right)$ be a field.

A valuation on $k$ is a map $\nu: k \to \Z \cup \left\{{+\infty}\right\}$ which has the following properties, for all $a,b \in k$:


 * $(1): \quad \nu \left({a \cdot b}\right) = \nu \left({a}\right) + \nu \left({b}\right)$
 * $(2): \quad \nu \left({a}\right) = +\infty \iff a = 0$
 * $(3): \quad \nu \left({a + b}\right) \ge \min \left\{{\nu \left({a}\right), \nu \left({b}\right) }\right\}$