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 polynomial with real coefficients has a real root

Statement

False claim: every polynomial with real coefficients has a real root.

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 $R$ ).

Refutation

technique · direct

1.1

Since $f$ is irreducible and has degree $2$ , it has no real root.

L1algebra

2.1

Thus $f$ is a real polynomial without any real zero.

step 1.1

3.1

This single counterexample refutes the claim that every real polynomial has a real root.

step 2.1∎

Depends on

$x^2+1$ is irreducible over $\mathbb R$

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

4 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

Thomas W. Judson, Abstract Algebra: Theory and Applications, extension fields (standard reference, not scraped)