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Let $L/K$ be an extension of fields, let $L_s$ be the separable closure of $K$ on $L$. Show that $L/L_s$ is purely inseparable and $L_s/ K$ is separable.

That $ L_s / K $ is separable, follows directly from the definition of $ L_s $. Could you help me show that $L /L_s$ is purely inseparable? Thanks..

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  • $\begingroup$ After that, the goal is to show $(F[x]/(f(x)))/F$ purely inseparable means $f' = 0$ thus $f(x) = g(x^p)$. $\endgroup$ – reuns Dec 8 '17 at 17:44
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Hint: Suppose some $\alpha\in L$ is separable over $L_s$. What can you say about $\alpha$ over $K$?

A full answer is hidden below.

Since $\alpha$ is separable over $L_s$ and $L_s$ is separable over $K$, $\alpha$ is separable over $K$. But that implies $\alpha\in L_s$ by definition of $L_s$. Thus any element of $L\setminus L_s$ is inseparable over $L_s$, so $L$ is purely inseparable over $L_s$.

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