Alphabeta Math
LemmaStatement: Literature-sourcedProof: Literature-sourcedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-31
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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 trace pairing in a finite separable extension is nondegenerate

Statement

Let L/F be a finite separable field extension. Then the bilinear pairing L×L→F, (x,y)↦Tr⁡L/F(xy), is nondegenerate.

Facts & Assumptions

Given: A finite separable extension L/F.

[L1]

The trace form of a finite extension is nondegenerate exactly when the extension is separable (The trace form of a finite extension is nondegenerate exactly when the extension is separable).

Proof

technique · direct
1.1

The displayed pairing is exactly the trace form of the extension L/F.

L1given
2.1

Because L/F is separable, [L1] makes that trace form nondegenerate.

L1step 1.1∎

Depends on

Used by

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