Lemma 10.46.6. Let $k'/k$ be a field extension. Let $p$ be a prime number. The following are equivalent
$k'$ is generated as a field extension of $k$ by elements $x$ such that there exists an $n > 0$ with $x^{p^ n} \in k$ and $p^ nx \in k$, and
$k = k'$ or the characteristic of $k$ and $k'$ is $p$ and $k'/k$ is purely inseparable.
Proof. Let $x \in k'$. If there exists an $n > 0$ with $x^{p^ n} \in k$ and $p^ nx \in k$ and if the characteristic is not $p$, then $x \in k$. If the characteristic is $p$, then we find $x^{p^ n} \in k$ and hence $x$ is purely inseparable over $k$. $\square$
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