Valuation ($v$)

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.

Two additive valuations are equivalent when they have the same valuation ring. Their ordered value-group images then identify compatibly with the valuations; for nontrivial real-valued rank-one valuations this is multiplication by a positive constant.

A nontrivial additive valuation has rank one when its ordered value group is Archimedean and therefore embeds order-preservingly in $\mathbb{R}$. Exponentiating a real-valued representative gives a Non-Archimedean absolute value; changing the embedding rescales the valuation positively. Higher-rank ordered value groups cannot be represented by real-valued absolute values on a field.

For a polynomial over a discretely valued field and a rational weight $t$, this coefficient-minimum function is multiplicative. Scale the variable by an element of valuation $t$ in a valued extension and reduce the initial terms; nonzero residue polynomials have nonzero product. For a monic polynomial factored into roots, $w_t(f)={\sum }_r\min (t,v(r))$. Its supporting lines determine the lower Newton polygon, giving the Newton polygon root valuation theorem and its multiplicities.

A non-Archimedean valuation satisfies the strong triangle inequalityIn multiplicative absolute-value notation this becomes $|x+y|\le \max \{|x|,|y|\}$.

An additive nontrivial valuation on the rational field is equivalent to exactly one p-adic valuation. Integers have nonnegative values; some prime must have positive value, and Bézout's identity prevents two distinct primes from doing so. Unique factorization then gives the formula, with $c>0$ in the real-valued case. The trivial valuation is a separate possibility if admitted by the definition.