ExampleConstruction: AI-generatedVerification: AI-adaptedPipeline-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.

A pure cubic power basis: why the squarefree-discriminant certificate does not apply

Example

Let $α = \sqrt[3]{2}$ and $K = Q (α)$ . The power basis $(1, α, α^{2})$ has discriminant $- 108$ . Consequently the squarefree power-basis criterion does not certify that $Z [α]$ is the full ring of integers. This example records the boundary of that criterion; it makes no claim here about the actual index of $Z [α]$ .

Facts & Assumptions

Given: $α = \sqrt[3]{2}$ and its polynomial $f (X) = X^{3} - 2$ .

[F1]

The discriminant of a power basis is the polynomial discriminant of the minimal polynomial (Power-basis and polynomial discriminants).

[F2]

The squarefree criterion concludes maximality only when the power-basis discriminant is squarefree (Squarefree power discriminant criterion).

Verification

technique · direct

1.1

Eisenstein's criterion at $2$ makes $f$ irreducible over $Q$ ; thus it is the monic minimal polynomial of the integral element $α$ and $(1, α, α^{2})$ is a $Q$ -basis of $K$ .

given

2.1

The cubic discriminant formula gives $disc (f) = - 27 (- 2)^{2} = - 108$ . By [F1] this is the discriminant of the displayed power basis.

F1step 1.1algebra

3.1

Since $- 108$ is divisible by $2^{2}$ and $3^{3}$ , it is not squarefree. Therefore the hypothesis of [F2] fails, so that criterion supplies no maximality conclusion in this example.

F2step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 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

William Stein, Algebraic Number Theory, Proposition 2.4.4 (standard reference, not scraped)