TheoremStatement: Literature-sourcedProof: Literature-sourcedjudge 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.

Power-basis and polynomial discriminants

Statement

Let $K = Q (α)$ . If $f$ is the degree- $n$ monic minimal polynomial of $α$ , then $disc (1, α, \dots, α^{n - 1}) = (- 1)^{n (n - 1) / 2} N_{K / Q} (f^{'} (α)) = disc (f) .$

Facts & Assumptions

Given: $K = Q (α)$ and $\deg f = n$ .

[F1]

The basis discriminant is the square of its embedding determinant (Embedding determinant formula).

Proof

technique · direct

1.1F1given

Fact [F1] makes the determinant the Vandermonde product of the conjugates of $α$ .

2.1step 1.1algebra∎

Squaring it and regrouping the root differences is the polynomial discriminant, equivalently the displayed derivative norm.

Depends on

Used by

Squarefree power discriminant criterion Corollary
Frobenius cycle type needs good reduction Counterexample
A pure cubic power basis: why the squarefree-discriminant certificate does not apply Example
Higher-degree class group by norm exclusions Example
Prime-power cyclotomic ring, discriminant support and p factor Lemma
Signed discriminant of a cyclotomic field Theorem

Dependency tree · two levels

8 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, Remark 2.28 (standard reference, not scraped)