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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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    Here's the file that EGA refers to for the proof (more precisely, it refers to pages 03-06): http://numdam.org/numdam-bin/fitem?id=SHC_1955-1956__8__A14_02014-09-01

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