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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