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At the end of Exposé I in SGA 1 it is asserted that a well known lemma states the following:

Let $k$ be an infinite field and $E$ be a finite product of finite field extensions of $k$. Suppose not all of them are separable extensions. Then there is an element of $E$ whose minimal polynomial has degree greater than the separable degree of $E/k$.

I think I know how to leverage the assumption that $k$ is infinite to reduce the problem to showing it when $E$ is a field. So let $E/k$ be a finite field extension ($k$ infinite but I dunno if you need that) of separable degree $m$ and degree $n$ with $m<n$. How to produce an element whose minimal polynomial has degree $> m$?

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