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?
Best Answer
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.