LemmaStatement: Literature-sourcedProof: Literature-sourcedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-11

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Every nonzero nonunit polynomial over a field factors into irreducible polynomials

Statement

Every nonzero nonunit polynomial over a field is a finite product of irreducible polynomials.

Facts & Assumptions

Given: A field $F$ and a nonzero nonunit polynomial $f\in F[x]$ .

[L1]

A nonzero nonunit is irreducible when every factorization has a unit factor (Irreducible and prime elements of an integral domain).

[L2]

Degrees add under multiplication of nonzero polynomials over a field (Over an integral domain, degrees add under multiplication of nonzero polynomials).

[L3]

The units of $F[x]$ are exactly its nonzero constant polynomials (The units of $R[x]$ over an integral domain are exactly the constant polynomials whose values are units of $R$ ).

[L4]

Strong induction proves a natural-number property once the case at $n$ follows from all smaller cases (Strong (complete) induction).

Proof

technique · induction

1.1

Use strong induction on $n=\deg f$ ; by [L3], a nonzero nonunit has $n\ge1$ .

basegivenL3L4

2.1

If $f$ is irreducible, it is already a one-factor product; otherwise [L1] gives $f=gh$ with $g,h$ nonunits, and neither is zero because $f\ne0$ .

step 1.1ihL1construct

3.1

By [L2], $\deg g$ and $\deg h$ are positive and strictly below $n$ , so the induction hypotheses factor both into irreducibles; concatenating those factorizations gives one for $f$ , and [L4] completes the induction.

step 2.1ihL2L3L4discharge-induction∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 39 results over 11 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

Neil Donaldson, Math 120B Notes, Theorem 23.7 (standard reference, not scraped)