Corollary 2

— Let $f \in K[X]$ be a non-zero polynomial. For $f$ to be separable it is necessary and sufficient that there should exist an extension $L$ of $K$ which is a perfect field and such that $f$ has no repeated factor in $L[X]$.

Let $\Omega$ be an algebraic closure of $K$; iff is separable, $f$ has no repeated factors in $\Omega[X]$ (Prop. 3, d)). Conversely, if $L$ is a perfect extension of $K$ such that $f$ has no repeated factor in $L[X]$, then $f$ is separable in $L[X]$ (Cor. 1 and Remark 2), hence in $K[X]$ (Remark 1).