Lemma 9.12.4. Let $F$ be a field. An irreducible polynomial $P$ over $F$ is separable if and only if $P$ has pairwise distinct roots in an algebraic closure of $F$.

**Proof.**
Suppose that $\alpha \in \overline{F}$ is a root of both $P$ and $P'$. Then $P = (x - \alpha )Q$ for some polynomial $Q$. Taking derivatives we obtain $P' = Q + (x - \alpha )Q'$. Thus $\alpha $ is a root of $Q$. Hence we see that if $P$ and $P'$ have a common root, then $P$ does not have pairwise distinct roots. Conversely, if $P$ has a repeated root, i.e., $(x - \alpha )^2$ divides $P$, then $\alpha $ is a root of both $P$ and $P'$. Combined with Lemma 9.11.2 this proves the lemma.
$\square$

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