CorollaryStatement: Literature-sourcedProof: AI-adaptedprecheck 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.

Every real polynomial factors into linear and irreducible quadratic factors

Statement

Every nonzero polynomial $f \in R [x]$ can be written as a nonzero real scalar times a product of linear polynomials and irreducible quadratic polynomials.

Facts & Assumptions

Given: A nonzero polynomial $f \in R [x]$ .

[L1]

Every nonzero nonunit polynomial over a field factors into irreducible polynomials (Every nonzero nonunit polynomial over a field factors into irreducible polynomials).

[L2]

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

Proof

technique · direct

1.1L1algebra

If $f$ is a nonzero constant, then it already has the required form: it is itself the nonzero scalar multiplying the empty product. Now assume that $f$ is a nonzero nonunit polynomial. By [L1], it factors as $f = u p_{1} \dots p_{r}$ with $u \in R^{\times}$ and each $p_{j}$ irreducible in $R [x]$ .

2.1L2step 1.1

By [L2], each irreducible factor $p_{j}$ has degree $1$ or $2$ . Therefore each $p_{j}$ is either linear or an irreducible quadratic, so the factorization of step 1.1 is exactly the required one.

3.1step 1.1step 2.1∎

The constant and nonconstant cases together prove the statement for every nonzero polynomial in $R [x]$ .

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