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An extension $L$ of a field $k$ is called primary if the relative algebraic closure of $k$ in $L$ is purely inseparable over $k$.

I'm looking for a proof of the following proposition which is used in EGA. It refers to the Cartan-Chevalley seminar, but I don't have an easy access to it.

EGA IV-2 (4.3.2) Let $K, L$ be extensions of a field $k$. Suppose $L$ is a primary extension of $k$. Then $Spec(L\otimes_k K)$ is irreducible and the residue field $\kappa(\xi)$ of the generic point $\xi$ of $Spec(L\otimes_k K)$ is a primary extension of $K$.

Conversely if $Spec(L\otimes_k K)$ is irreducible for every finite separable extension $K$ of $k$, $L$ is a primary extension of $k$.

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If no one will post a solution, I can give you some hints coming from the Cartan-Chevalley seminar. –  user26857 Dec 28 '12 at 17:21
    
@YACP Thanks. Could you give us a full proof? I think the question is interesting and important not only for me. –  Makoto Kato Dec 28 '12 at 18:27
    
I don't have a proof of my own (actually I'm not interested in this, I simply wanted to help you) and the original proof of Cartier seems to be long enough. But let's wait for a while, maybe someone will post an answer. –  user26857 Dec 28 '12 at 19:09
    
@YACP OK. Let's wait for a while. And if no one would post an answer, please give us hints. Thank you. –  Makoto Kato Dec 28 '12 at 19:13

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