Valuation on a field

Definition

Let $K$ be a field and let $\Gamma$ be an ordered abelian group, written additively. A valuation on $K$ with values in $\Gamma$ is a map

v:K\longrightarrow\Gamma\cup\{\infty\}

such that

v(0)=\infty,\qquad v(1)=0,\qquad v(xy)=v(x)+v(y),\qquad v(x+y)\ge\min\{v(x),v(y)\}.

Here $\infty$ is larger than every element of $\Gamma$ and $\infty+\gamma=\infty$ . The pair $(K,v)$ is a valued field.

Dominance and cancellation

If $v(x)\ne v(y)$ , then the ultrametric inequality forces

v(x+y)=\min\{v(x),v(y)\}.

When $v(x)=v(y)$ , equality may fail because leading terms can cancel, raising the valuation. This unequal-versus-tied dichotomy is exactly what the tropical hyperfield records.

Valuation ring and residue field

The valuation ring and its maximal ideal are

\mathcal O_v=\{x\in K:v(x)\ge0\},\qquad \mathfrak m_v=\{x\in K:v(x)>0\}

respectively. The quotient $\kappa(v)=\mathcal O_v/\mathfrak m_v$ is the residue field. The subgroup $v(K^\times)$ is the value group; one may replace $\Gamma$ by this subgroup when a surjective valuation is desired.

Additive and multiplicative conventions

The $p$ -adic valuation is additive as above. A non-Archimedean absolute value is the multiplicative-size convention, satisfying $|x+y|\le\max(|x|,|y|)$ . For real-valued $v$ , a typical conversion is $|x|=e^{-v(x)}$ . Thus larger additive valuation means smaller absolute value, and formulas must not mix the two order conventions.

References

Irving Kaplansky, “Maximal fields with valuations,” Duke Mathematical Journal 9 (1942), 303–321. Project Euclid DOI record. Relevant: valued fields and value groups.
Oleg Viro, “Hyperfields for Tropical Geometry I: Hyperfields and dequantization,” 2010. arXiv:1006.3034. Relevant: non-Archimedean norms and tropical hyperfields.