LemmaStatement: Literature-sourcedProof: Literature-sourcedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-06

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.

Change of basis for discriminants

Statement

If $β_{j} = \sum_{i} a_{i j} α_{i}$ , then $disc (β) = \det (A)^{2} disc (α)$ .

Facts & Assumptions

Given: Two ordered bases related by $A = (a_{i j})$ .

Proof

technique · direct

1.1

Bilinearity of trace gives $G_{β} = A^{t} G_{α} A$ for the trace Gram matrices.

givenalgebra

2.1

Taking determinants gives the asserted square factor.

step 1.1algebra∎

Depends on

Discriminant of a basis and order

Used by

Order-index discriminant formula Corollary
Number-field discriminant is well-defined and nonzero Theorem

Cited to discharge well-definedness by Discriminant of a basis and order.

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

Milne, Lemma 2.23 (standard reference, not scraped)