DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-08-31

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Valuations on a field

Definition

Let $K$ be a field and let $Γ$ be a totally ordered abelian group (Totally ordered abelian groups). Adjoin a symbol $\infty$ with $γ < \infty, γ + \infty = \infty + γ = \infty, \infty + \infty = \infty .$ for every $γ \in Γ$ .

A valuation on $K$ with value group in $Γ$ is a map $v : K \to Γ \cup {\infty}$ such that for all $x, y \in K$ , $v (x) = \infty ⟺ x = 0,$ $v (x y) = v (x) + v (y),$ and $v (x + y) \geq \min {v (x), v (y)} .$

When $x \neq 0$ , the value $v (x)$ lies in $Γ$ . The displayed laws imply $v (1) = 0$ and $v (x^{- 1}) = - v (x)$ for $x \in K^{\times}$ .

Depends on

Totally ordered abelian groups

Used by

Two DVR lifts of one diagram over the doubled-origin line Counterexample
Discrete valuations Definition
A nondiscrete rank-one valuation from incommensurate values Example
A rank-two valuation ring that is not a DVR Example
dvrs as regular local rings Example
Proper normal curve rational function map Lemma
Valuation lifts detect universal closedness Lemma
A valuation ring is recovered from its value group Theorem
Cartier divisors on a normal Noetherian scheme give Weil divisors Theorem

Dependency tree · one level

1 result within one dependency step of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

M. Mustata, Commutative Algebra, Definition 8.1 and Remark 8.2 (standard reference, not scraped)
Allen B. Altman and Steven L. Kleiman, A Term of Commutative Algebra, 13th ed., (26.11) (standard reference, not scraped)