ExampleConstruction: 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 in a cubic field

Example

For $f = X^{3} - X - 1$ , modulo $2$ one has $\overset{ˉ}{f} = X^{3} + X + 1$ , irreducible; whenever the index hypothesis holds, $(2)$ is inert in $Q (α)$ .

Verification

Given: a root $α$ with $O_{K} = Z [α]$ .

1.1

Neither $0$ nor $1$ is a root of $X^{3} + X + 1$ in $F_{2}$ , so the cubic is irreducible.

givenalgebra

2.1

Dedekind--Kummer then gives one prime above $2$ with residue degree $3$ and exponent $1$ , which is inertness.

step 1.1algebra∎

Depends on

Dedekind--Kummer prime factorisation

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

J. S. Milne, Algebraic Number Theory, Example 3.48 (standard reference, not scraped)