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In the proof of Corollary 10.12 of Exposé I of SGA 1 something like the following is asserted as a ‘known lemma’:

Let $k$ be an infinite field and $B$ a finite $k$-algebra. If $B$ is not a product of separable field extensions of $k$, then there exists an element of $B$ whose minimal polynomial has degree greater than the separable degree of $B$ (= sum of separable degrees of the residue fields of $B$).

Where can I find a proof of this lemma? I don't even see it when $B$ is a field.

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    $\begingroup$ Corollary 10.12 there is a special case of EGA IV$_4$ 18.10.16 and the latter has a complete proof written, so look in that proof and ignore the proof given in SGA1 (the same advice can be applied to many things in the early exposes of SGA1, keeping in mind that EGA was among other things meant to provide a more complete treatment of that "introductory" material). $\endgroup$
    – nfdc23
    Commented Oct 2, 2017 at 2:36

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