False statementConstruction: AI-adaptedVerification: AI-generatedprecheck passjudge pass (deepseek-v4-pro + claude-sonnet-5)audited 2026-08-17

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.

FALSE: every irreducible polynomial over a field is separable

Statement

Every irreducible polynomial over every field is separable.

Facts & Assumptions

Given: The universal claim in the Statement.

[L1]

Over $F_{p} (t)$ , the polynomial $x^{p} - t$ is irreducible, has zero derivative, and is inseparable ( $x^{p} - t$ is irreducible and inseparable over $F_{p} (t)$ ).

Refutation

technique · direct

1.1L1

The polynomial in [L1] is an irreducible polynomial over a field which is not separable.

2.1step 1.1∎

It is a counterexample to the universal claim, so the Statement is false.

Depends on

$x^p-t$ is irreducible and inseparable over $\mathbb F_p(t)$

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

P. L. Clark, Field Theory, Chapter 4 (standard reference, not scraped)
J. S. Milne, Fields and Galois Theory, Chapters 3 and 5 (standard reference, not scraped)