Any algebraic field extension is uniquely a separable field extension followed by a purely inseparable one.

Lemma 9.14.6. Let $E/F$ be an algebraic field extension. There exists a unique subextension $F \subset E_{sep} \subset E$ such that $E_{sep}/F$ is separable and $E/E_{sep}$ is purely inseparable.

Proof. If the characteristic is zero we set $E_{sep} = E$. Assume the characteristic if $p > 0$. Let $E_{sep}$ be the set of elements of $E$ which are separable over $F$. This is a subextension by Lemma 9.12.13 and of course $E_{sep}$ is separable over $F$. Given an $\alpha$ in $E$ there exists a $p$-power $q$ such that $\alpha ^ q$ is separable over $F$. Namely, $q$ is that power of $p$ such that the minimal polynomial of $\alpha$ is of the form $P(x^ q)$ with $P$ separable algebraic, see Lemma 9.12.1. Hence $E/E_{sep}$ is purely inseparable. Uniqueness is clear. $\square$

Comment #826 by on

Suggested slogan: Algebraic field extensions break down uniquely into a separable and purely inseparable part.

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