Lemma 9.14.4 (09HH)—The Stacks project

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Lemma 9.14.4. Let $E/k$ be a field extension. Then the elements of $E$ purely-inseparable over $k$ form a subextension of $E/k$ .

Proof. Let $p$ be the characteristic of $k$ . Let $\alpha , \beta \in E$ be purely inseparable over $k$ . Say $\alpha ^ q \in k$ and $\beta ^{q'} \in k$ for some $p$ -powers $q, q'$ . If $q''$ is a $p$ -power, then $(\alpha + \beta )^{q''} = \alpha ^{q''} + \beta ^{q''}$ . Hence if $q'' \geq q, q'$ , then we conclude that $\alpha + \beta$ is purely inseparable over $k$ . Similarly for the difference, product and quotient of $\alpha$ and $\beta$ . $\square$

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