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

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A polynomial of degree two or three over a field is irreducible exactly when it has no root in the field

Statement

Let $F$ be a field and let $f\in F[x]$ have degree $2$ or $3$ . Then $f$ is irreducible over $F$ if and only if $f$ has no root in $F$ .

Facts & Assumptions

Given: A field $F$ and a polynomial $f\in F[x]$ of degree $2$ or $3$ .

[L1]

An element $a\in F$ is a root of $f$ exactly when $x-a$ divides $f$ (Factor theorem over a commutative ring).

[L2]

Degrees add in a product 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 the nonzero constants (The units of $R[x]$ over an integral domain are exactly the constant polynomials whose values are units of $R$ ).

[L4]

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

Proof

technique · direct

1.1

If $f$ has a root $a$ , then [L1] gives $f=(x-a)q$ ; [L2] and $\deg f\ge2$ make both factors nonunits by [L3], so [L4] shows that $f$ is reducible.

givenL1L2L3L4

2.1

Conversely, if $f=gh$ with both factors nonunits, [L2] and [L3] give positive degrees summing to $2$ or $3$ , so one factor has degree $1$ ; writing it as $cx+d$ with $c\ne0$ , it has root $-c^{-1}d$ , and that root is a root of $f$ . Thus reducibility implies a root, proving the biconditional.

givenL2L3L4algebra∎

Depends on

Used by

Reduction modulo 2 proves x³+17x+391 irreducible over ℚ Example
The polynomial x²-2 is irreducible over ℚ Example
The polynomial x²+1 is irreducible over ℝ Example

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 25 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 23.8 (standard reference, not scraped)