Separable polynomial

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.