Proposition 4

— Let $f$ be an irreducible polynomial in $K[X]$. Then the following conditions are equivalent:

a) $f$ is separable. b) There exists an extension $L$ of $K$ in which $f$ has a simple root. c) The derivative $f'$ off is not zero. d) The field $K$ is of characteristic $0$, or it is of characteristic $p \neq 0$ and $f \not\in K[X^p]$.

We note first that an irreducible polynomial in $K[X]$ is not constant. It is clear that a) implies b) (take an algebraic closure of $K$ for $L$). If $x$ is a simple root off in an extension $L$ of $K$, we have $f'(x) \neq 0$ (IV, p. 17, Prop. 7), so b) implies c), and the equivalence of c) and d) follows from V, p. 9, Cor.

Suppose finally that $f \neq 0$; let $x$ be a root of $f$ in an algebraically closed extension $\Omega$ of $K$. Since $f$ is the minimal polynomial of $x$ over $K$ and $\deg f' < \deg f$, we have $f'(x) \neq 0$, and so $x$ is a simple root of $f$ (IV, p. 17, Prop. 7). Therefore $f$ is separable and we have shown that c) implies a).