CounterexampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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.

Dedekind--Kummer without the index hypothesis

Statement refuted

Reduction of a minimal polynomial modulo $p$ always gives the prime factorisation of $p O_{K}$ .

Counterexample

Given: $K = Q (\sqrt{5})$ and $α = \sqrt{5}$ .

1.1

The order $Z [\sqrt{5}]$ has discriminant $20$ , whereas $d_{K} = 5$ , so the index formula gives index $2$ .

givenalgebra

2.1

Yet $X^{2} - 5 \equiv (X + 1)^{2} (m o d 2)$ while $2$ is unramified because $2 ∤ d_{K}$ . Thus the repeated reduction factor incorrectly predicts ramification, demonstrating necessity of the index hypothesis.

step 1.1algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

9 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

J. S. Milne, Algebraic Number Theory, Remark 3.42 (standard reference, not scraped)