Corollary 1

— For a field $K$ to be perfect it is necessary and sufficient that every irreducible polynomial of $K[X]$ should be separable.

If the field $K$ is of characteristic $0$, $K$ is perfect and every irreducible polynomial of $K[X]$ is separable, by d) above. Suppose then that $K$ has characteristic $p \neq 0$.

Suppose first that $K$ is perfect. We have $K[X^p] = K[X]^p$, hence there exists no irreducible polynomial of $K[X]$ belonging to $K[X^p]$. By Prop. 4, every irreducible polynomial of $K[X]$ is then separable.

Suppose next that $K$ is imperfect, whence $K \neq K^p$. Let $a$ be an element of $K$ not belonging to $K^p$; the polynomial $X^p - a$ is irreducible in $K[X]$ (V, p. 24, Lemma 1), and it belongs to $K[X^p]$, and so is not separable.