Lemma 10.45.2. A field $k$ is perfect if and only if it is a field of characteristic $0$ or a field of characteristic $p > 0$ such that every element has a $p$th root.

Proof. The characteristic zero case is clear. Assume the characteristic of $k$ is $p > 0$. If $k$ is perfect, then all the field extensions where we adjoin a $p$th root of an element of $k$ have to be trivial, hence every element of $k$ has a $p$th root. Conversely if every element has a $p$th root, then $k = k^{1/p}$ and every field extension of $k$ is separable by Lemma 10.44.1. $\square$

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