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

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

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

Valuations on a field

Definition

Depends on

Dependency tree · 1 level

Sources