LemmaStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-05

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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.

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The Hilbert symbol depends only on square classes

Statement

For $a, b, u, v \in Q_{w}^{\times}$ ,

(a, b)_{w} = (a u^{2}, b v^{2})_{w} .

In particular, the Hilbert symbol depends only on the square classes of its two arguments.

Facts & Assumptions

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

[L1]

By definition, $(c, d)_{w} = 1$ exactly when $c x^{2} + d y^{2} = 1$ is solvable over $Q_{w}$ (The Hilbert symbol over a rational completion).

Proof

technique · direct

1.1

If $(a, b)_{w} = 1$ , choose $x, y$ with $a x^{2} + b y^{2} = 1$ by [L1]. Then $(x u^{- 1}, y v^{- 1})$ solves $(a u^{2}) X^{2} + (b v^{2}) Y^{2} = 1$ , so $(a u^{2}, b v^{2})_{w} = 1$ .

L1givenalgebra

2.1

The same argument with $u^{- 1}$ and $v^{- 1}$ shows that if $(a u^{2}, b v^{2})_{w} = 1$ , then $(a, b)_{w} = 1$ . Hence the two symbols are equal, and only the square classes matter.

L1step 1.1algebra∎

Depends on

The Hilbert symbol over a rational completion

Used by

The Hilbert symbol is a symmetric bilinear nondegenerate pairing Theorem
The odd-prime Hilbert symbol formula Theorem
The two-adic Hilbert symbol formula Theorem

Dependency tree · two levels

2 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, section 4 (standard reference, not scraped)