DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-05

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.

The rational local fields

Definition

The rational local fields are the completions of $Q$ at its places:

$Q_{\infty} : = R$ at the archimedean place,
$Q_{p} : = Q_{p}$ at the prime place $p$ , where $Q_{p}$ is the completion of The p-adic numbers as a metric completion, equipped with its field structure by The p-adic completion is a complete valued field.

When a statement is uniform in the place $v$ , the notation $Q_{v}$ refers to one of these fields.

Depends on

Used by

The Hilbert symbol over a rational completion Definition

Dependency tree · two levels

6 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

Andrew V. Sutherland, 18.782 Lecture 10 (standard reference, not scraped)
Sam Raskin, Introduction to the Arithmetic Theory of Quadratic Forms (standard reference, not scraped)