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 Hilbert symbol over a rational completion

Definition

Let $v \leq \infty$ be a place of $Q$ , and let $a, b \in Q_{v}^{\times}$ . The Hilbert symbol $(a, b)_{v}$ is defined by

$(a, b)_{v} = {\begin{cases} 1, & if a x^{2} + b y^{2} = 1 has a solution in Q_{v}, \\ - 1, & otherwise. \end{cases}$

The variables $x, y$ are allowed to be any elements of $Q_{v}$ . The next lemma recasts the same condition as isotropy of a ternary form and as a norm condition from the quadratic algebra $Q_{v} [T] / (T^{2} - a)$ , which is a quadratic field extension when $a$ is nonsquare and a split algebra when $a$ is square.

Depends on

The rational local fields

Used by

Dependency tree · two levels

3 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, Definition 10.1 (standard reference, not scraped)
Sam Raskin, Introduction to the Arithmetic Theory of Quadratic Forms, section 4 (standard reference, not scraped)