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Let $K$ be an Henselian discrete valuation field such that its completion is separable over $K$. Let $F$ be its infinite residue field. Is it true that a finite extension of $F$ is a simple extension i.e., it is of the form $F(\alpha)$ for some $\alpha$?

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A finite extension of any field is simple, by the primitive element theorem. –  Will Sawin Apr 6 '13 at 5:10
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@Sawin: It is true only for finite separable extensions. –  Jana Apr 6 '13 at 5:38
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Maybe I misunderstand the question, but what if you start with an extension $L/F$ which is not simple and then take $K=F((t))$? –  Laurent Moret-Bailly Apr 6 '13 at 6:23
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