Separable polynomial

Definition

Let $R$ be a commutative ring and let $f \in R[x]$. The polynomial $f$ is separable if there are $a, b \in R[x]$ with $a f + b f' = 1$, where $f'$ is the formal derivative of $f$.

Remarks

Over a field $K$, $f$ is separable if and only if $f$ has no repeated root in an algebraic closure of $K$. The polynomial $f = x^2 - 2$ is separable over $\mathbb{Q}$, because $f' = 2x$ and $-\tfrac{1}{2} f + \tfrac{1}{4} x f' = 1$. Over $\mathbb{F}_p(t)$, the irreducible polynomial $x^p - t$ is not separable, because its derivative is $0$. Some older texts call an irreducible polynomial separable when it has distinct roots, and extend this to other polynomials factor by factor.

Explanations

L1 human

A separable polynomial has distinct roots, in a field that is large enough to contain all of them. Over the rational numbers, every irreducible polynomial is separable. Inseparable irreducible polynomials only appear in prime characteristic.