Valuation ($v$)

A valuation assigns sizes to elements of a field in a way compatible with multiplication and addition. In additive notation, $v(xy)=v(x)+v(y)$ and $v(x+y)\ge \min \{v(x),v(y)\}$.

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.