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I am a bit puzzled by the following statement in Serre's local algebra page 46, in the proof of Proposition 16 (https://books.google.com.au/books?id=oZmXsvlYtMQC&pg=PA46&source=gbs_toc_r&cad=3#v=onepage&q&f=false). Let me isolate the statement:

Let $K \leq L$ be a finite field extension. By $M \subset L$ we denote the set of elements that are purely inseparable over $K$, i.e., $v \in M$ if and only if $v^{p^r} \in K$ for some $r > 0$. It is not hard to see that $M$ is a field, extending $K$. Then it is claimed that $M \leq L$ is separable ?!

This puzzles, me as I know that one can find $K_{sep}$ such that $K \leq K_{sep}$ is separable and $K \leq L$ is purely inseparable, however I suspect the above can not be true, in fact I vaguely remember counterexamples existing. Can someone help me out about what Serre means here and what I am missing?

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  • $\begingroup$ Your $E$ seems to appear out of nowhere. Is this meant to be $M$? $\endgroup$ Nov 12 '19 at 1:28
  • $\begingroup$ It is indeed. Fixed. $\endgroup$
    – Hammerhead
    Nov 12 '19 at 1:29
  • $\begingroup$ I vaguely remember something like that, too. I’ll be interested to see what an expert says about this. $\endgroup$
    – Lubin
    Nov 12 '19 at 4:13
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It really does seem to be wrong. There are examples in an earlier MSE posting. Your question boils down to this: if, in an extension $L\supset K$, the field $L$ has no elements purely inseparable over $K$, is $L$ separable over $K$? And the answer, grâce à Joe Lipman, is no.

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  • $\begingroup$ Thanks. In that case, is there any way to make Serre's argument work? $\endgroup$
    – Hammerhead
    Nov 12 '19 at 15:39
  • $\begingroup$ I don’t see any way around the problem. Maybe if the original extension is normal? And note that the two examples there are in characteristic $2$. One should check that a similar construction works in any positive characteristic. But I’m feeling too lazy. $\endgroup$
    – Lubin
    Nov 12 '19 at 22:28

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