False statementConstruction: AI-adaptedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-27

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 in $R [x]$ has degree $1$

Statement

False claim: every irreducible polynomial in $R [x]$ has degree $1$ .

Facts & Assumptions

Given: The polynomial $f (x) = x^{2} + 1$ .

[L1]

The polynomial $x^{2} + 1$ is irreducible in $R [x]$ ( $x^{2} + 1$ is irreducible over $R$ ).

[L2]

An irreducible polynomial in $R [x]$ has degree $1$ or $2$ (An irreducible polynomial in $R [x]$ has degree $1$ or $2$ ).

Refutation

technique · direct

1.1

By [L1], the polynomial $x^{2} + 1$ is irreducible over $R$ .

2.1

It has degree $2$ , and [L2] says that degree $2$ is the exact non-linear possibility for irreducible real polynomials.

L2step 1.1

3.1

Therefore the degree-one claim is false.

step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

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

J. S. Milne, Fields and Galois Theory, v5.10, Chapter 5 (standard reference, not scraped)

FALSE: every irreducible polynomial in R[x]R[x] has degree 11