 Valuation

An additive valuation on a field $K$ is a map $v:K\to \Gamma \cup \{\infty \}$, where $\Gamma$ is a totally ordered group with commutative addition, satisfyingThe order is compatible with addition, and $\infty$ exceeds every element of $\Gamma$. One usually takes the value group to be the image of $K^{\times }$. The trivial valuation assigns zero to every nonzero element.

For a prime number $p$, writing a nonzero rational number as $p^{r}a/b$ with $p$ dividing neither $a$ nor $b$ gives the p-adic valuation $v_p(p^{r}a/b)=r$. Thus $v_2(12/5)=2$. Products add exponents, whereas cancellation in a sum can only increase the minimum exponent. In fact, if $v(x)\ne v(y)$ then $v(x+y)=\min \{v(x),v(y)\}$: applying the inequality to $x=(x+y)-y$ rules out a strict increase. The nonnegative elements form the valuation ring, whose positive-valued elements form its maximal ideal.