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...where $K^{1/p^\infty}$ is the perfect closure of $K$.

Discussed with my classmates and we think this is not necessarily true, so we want to find a counterexample, where $E$ is an inseparable but not purely inseparable extension of $E\cap K^{1/p^\infty}$.

To show some of my thoughts: Maybe because of the examples used in the lecture, I kept thinking of examples like simple extensions $K(\alpha)$ where $\alpha$ is a root of $x^n-t$, and $K$ is the infinite field of characteristic $p$, for example $\mathbb{Z}_2(t)$. But later on I thought in general this kind of thing does not work. Since if $n=mp^k$, where $p\nmid m$, then $K(\alpha^m)$ would be purely inseparable over $K$, but $K(\alpha)$, will be separable over $K(\alpha^m)$. Therefore, if thinking of simple extensions, we need some more complicated minimal polynomials.

Just cannot get rid of thinking along this way...

Thanks in advance for any help!!

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