The Stacks project

Lemma 9.28.2. Let $K$ be a field of characteristic $p > 0$. Let $L/K$ be a separable algebraic extension. Let $\alpha \in L$.

  1. 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 $L$.

  2. 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 $L$.

Proof. It follows from the definitions that (2) implies (1). Assume $P$ is as in (2). Write $P(T) = \sum \nolimits _{i = 0}^ d a_ i T^{d - i}$ and $a_ i = b_ i^ p$. The polynomial $Q(T) = \sum \nolimits _{i = 0}^ d b_ i T^{d - i}$ has distinct roots in an algebraic closure as well, because the roots of $Q$ are the $p$th roots of the roots of $P$. If $\alpha $ is not a $p$th power, then $T^ p - \alpha $ is an irreducible polynomial over $L$ (Lemma 9.14.2). Moreover $Q$ and $T^ p - \alpha $ have a root in common in an algebraic closure $\overline{L}$. Thus $Q$ and $T^ p - \alpha $ are not relatively prime, which implies $T^ p - \alpha | Q$ in $L[T]$. This contradicts the fact that the roots of $Q$ are pairwise distinct. $\square$

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