Root in algebraic closure

Thanks to xd
https://stacks.math.columbia.edu/tag/09H3#comment-2963

fields.tex

@@ -1180,7 +1180,7 @@ \section{Separable extensions}
 \end{lemma}
 
 \begin{proof}
-Suppose that $\alpha \in F$ is a root of both $P$ and $P'$.
+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 $P$