LemmaStatement: Literature-sourcedProof: AI-generatedprecheck passjudge 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.

Binary quadratic representation via the Hilbert symbol

Statement

Let $a, b, t \in Q_{v}^{\times}$ . Then the binary form $a X^{2} + b Y^{2}$ represents $t$ over $Q_{v}$ if and only if

(a t, b t)_{v} = 1.

Facts & Assumptions

Given: A place $v$ of $Q$ and nonzero elements $a, b, t \in Q_{v}$ .

[L1]

The Hilbert symbol satisfies $(c, d)_{v} = 1$ exactly when $c X^{2} + d Y^{2} = 1$ is soluble over $Q_{v}$ (The Hilbert symbol over a rational completion).

[L2]

The symbol depends only on square classes (The Hilbert symbol is a symmetric bilinear nondegenerate pairing).

Proof

technique · direct

1.1

If $a X^{2} + b Y^{2} = t$ has a solution, divide by $t$ to obtain $(a / t) X^{2} + (b / t) Y^{2} = 1$ . By [L1], this means $(a / t, b / t)_{v} = 1$ . Since $(a / t) / (a t) = t^{- 2}$ and $(b / t) / (b t) = t^{- 2}$ are squares, [L2] gives $(a t, b t)_{v} = 1$ .

L1L2givenalgebra

2.1

Conversely, if $(a t, b t)_{v} = 1$ , then [L2] gives $(a / t, b / t)_{v} = 1$ , because the two pairs differ by multiplying each entry by the same square $t^{- 2}$ . By [L1], there exist $X, Y$ with $(a / t) X^{2} + (b / t) Y^{2} = 1$ , and multiplying by $t$ yields $a X^{2} + b Y^{2} = t$ .

L1L2step 1.1algebra∎

Depends on

Used by

Ternary isotropy via the Hilbert symbol Corollary

Dependency tree · two levels

9 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

Sam Raskin, Introduction to the Arithmetic Theory of Quadratic Forms, section 4.4 (standard reference, not scraped)
Andrew V. Sutherland, 18.782 Lecture 10 (standard reference, not scraped)