# largest purely inseparable extension (Serre, Local algebra p.46)

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?

• Your $E$ seems to appear out of nowhere. Is this meant to be $M$? Nov 12, 2019 at 1:28
• It is indeed. Fixed. Nov 12, 2019 at 1:29
• I vaguely remember something like that, too. I’ll be interested to see what an expert says about this. Nov 12, 2019 at 4:13

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.
• 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. Nov 12, 2019 at 22:28