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Let $L|K$ be a finite separable extension of fields complete under some valuation and let $\lambda, \kappa$ be residue class fields of $L$ and $K$ respectively. I guess that we do not know $\lambda|\kappa$ is not separable in general though it is hard for me to come up with a counter example. So I want to ask for an example when $\lambda|\kappa$ is inseparable and also if there is a natural condition to deduce that $\lambda|\kappa$ is separable.

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Most cases in "classical" number theory have perfect residue fields and so separable residue field extensions. But there are many examples of inseparable residue field extensions. Search for tamely ramified / wildy ramified and you will get a lot of information on your question.

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