Lemma 9.28.2. Let $K$ be a field of characteristic $p > 0$. Let $K \subset L$ be a separable algebraic extension. Let $\alpha \in L$.
If the coefficients of the minimal polynomial of $\alpha $ over $K$ are $p$th powers in $K$ then $\alpha $ is a $p$th power in $L$.
More generally, if $P \in K[T]$ is a polynomial such that (a) $\alpha $ is a root of $P$, (b) $P$ has pairwise distinct roots in an algebraic closure, and (c) all coefficients of $P$ are $p$th powers, then $\alpha $ is a $p$th power in $L$.