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.

Discrete valuation rings

Definition

Let $K$ be a field and let $v$ be a discrete valuation on $K$ (Discrete valuations). Its valuation ring is $V_{v} : = {x \in K : v (x) \geq 0} .$ This is a valuation ring in the sense of Valuation rings.

A discrete valuation ring (DVR) is a subring $V \subseteq K$ of the form $V = V_{v}$ for some discrete valuation $v$ on $K$ . Because $v$ is surjective, there is an element of value $1$ , so $V$ has a nonunit and is therefore not a field.

Discrete valuation rings

Definition

Depends on

Dependency tree · 2 levels

Sources