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

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An irreducible polynomial over a field is separable exactly when its derivative is nonzero

Statement

Let $F$ be a field and let $p\in F[x]$ be irreducible. Then $p$ is separable if and only if $p'\ne0$ .

Facts & Assumptions

Given: A field $F$ and an irreducible polynomial $p\in F[x]$ .

[L1]

A nonzero polynomial is separable exactly when its monic gcd with its derivative is $1$ (A nonzero polynomial over a field is separable exactly when its gcd with its derivative is $1$ ).

[L2]

If $p'\ne0$ , then $\deg p'\le\deg p-1$ (Linearity, power rule, Leibniz rule and the degree bound for the formal derivative).

[L3]

An irreducible polynomial is a nonzero nonunit whose only divisors are units and associates (Irreducible and prime elements of an integral domain).

Proof

technique · direct

1.1

If $p'\ne0$ , any common divisor of $p$ and $p'$ is a unit: a nonunit divisor of irreducible $p$ would be associate to $p$ by [L3], contradicting the strict degree bound [L2]; hence $\gcd(p,p')=1$ and [L1] makes $p$ separable.

givenL1L2L3

2.1

If $p'=0$ , then the monic associate of $p$ is the nonconstant gcd of $p$ and $0$ , so [L1] says that $p$ is not separable; this proves the converse and the biconditional.

givenL1L3∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 44 results over 12 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

Brian Conrad, Differential Criterion and Primitivity, Section 1 (standard reference, not scraped)