The Stacks project

Lemma 33.36.9. Let $k$ be a field of characteristic $p > 0$. Let $X$ be a scheme over $k$. Then $X$ is geometrically reduced if and only if $X^{(p)}$ is reduced.

Proof. Consider the absolute frobenius $F_ k : k \to k$. Then $F_ k(k) = k^ p$ in other words, $F_ k : k \to k$ is isomorphic to the embedding of $k$ into $k^{1/p}$. Thus the lemma follows from Lemma 33.6.4. $\square$

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