= Valuation
{title2=$v$}
{wiki=Valuation_(algebra)}
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>, satisfying
$$
v(0)=\infty,\qquad v(x)\in\Gamma\ (x\ne0),\qquad v(xy)=v(x)+v(y),\qquad v(x+y)\geq\min\{v(x),v(y)\}.
$$
The 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>.
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