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Let $K$ be a local field in the French sense, i.e. any field complete with respect to a discrete valuation, and suppose that the residue field of $K$ is perfect of characteristic $p>0$. What are examples of such $K$ that are not simply finite extensions of $\mathbb{Q}_p$?

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For any perfect field $k$ of char. $p$, you can consider the Witt vectors $W(k)$ of $k$, and then let $K$ be the fraction field of $W(k)$.

E.g. if $k = \overline{\mathbb F}_p$, then this gives the completion of the maximal unramified extension of $\mathbb Q_p$.

Then taking finite extension of this field (obtained say by adding $p^{1/n}$, or perhaps $p$-power roots of $1$) gives further examples.

These examples are still completions of (infinite) algebraic extensions of $\mathbb Q_p$. To get bigger examples, you could take $k$ to the be perfect closure of $\mathbb F_p(x)$, and then consider the fraction field of its ring of Witt vectors.

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