DefinitionDefinition: AI-adaptedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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.

Completion of a number field at a prime

Definition

For a nonzero prime P of $O_{K}$ , let $K_{P}$ be the completion at $∣ x ∣_{P} = (N P)^{- {ord}_{P} x}$ . Its valuation ring has residue field $O_{K} / P$ , since the original valuation ring is $(O_{K})_{P}$ and completion preserves residues. In L/K with $P ∣ p$ , the normalized value restricts as $∣ \cdot ∣_{P} ∣_{K} = ∣ \cdot ∣_{p}^{e f}$ , because $N P = (N p)^{f}$ and ${ord}_{P} ∣_{K} = e {ord}_{p}$ . When a literal extension of $∣ \cdot ∣_{p}$ is needed use $∣ \cdot ∣_{P}^{1 / (e f)}$ . Positive powers define the same topology and completion.

Depends on

Used by

Number field completions as local polynomial factors Lemma

Dependency tree · two levels

19 results within two dependency steps 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

Chapter 7, Theorem 7.38 and Remark 7.39; Chapter 8, Proposition 8.2 (standard reference, not scraped)