Lemma 9.14.5. Let $E/F$ be a finite purely inseparable field extension of characteristic $p > 0$. Then there exists a sequence of elements $\alpha _1, \ldots , \alpha _ n \in E$ such that we obtain a tower of fields

$E = F(\alpha _1, \ldots , \alpha _ n) \supset F(\alpha _1, \ldots , \alpha _{n - 1}) \supset \ldots \supset F(\alpha _1) \supset F$

such that each intermediate extension is of degree $p$ and comes from adjoining a $p$th root. Namely, $\alpha _ i^ p \in F(\alpha _1, \ldots , \alpha _{i - 1})$ is an element which does not have a $p$th root in $F(\alpha _1, \ldots , \alpha _{i - 1})$ for $i = 1, \ldots , n$.

Proof. By induction on the degree of $E/F$. If the degree of the extension is $1$ then the result is clear (with $n = 0$). If not, then choose $\alpha \in E$, $\alpha \not\in F$. Say $\alpha ^{p^ r} \in F$ for some $r > 0$. Pick $r$ minimal and replace $\alpha$ by $\alpha ^{p^{r - 1}}$. Then $\alpha \not\in F$, but $\alpha ^ p \in F$. Then $t = \alpha ^ p$ is not a $p$th power in $F$ (because that would imply $\alpha \in F$, see Lemma 9.12.5 or its proof). Thus $F \subset F(\alpha )$ is a subextension of degree $p$ (Lemma 9.14.2). By induction we find $\alpha _1, \ldots , \alpha _ n \in E$ generating $E/F(\alpha )$ satisfying the conclusions of the lemma. The sequence $\alpha , \alpha _1, \ldots , \alpha _ n$ does the job for the extension $E/F$. $\square$

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