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Let $K$ be a local field of positive characteristic. I'm looking for a $K$ which satisfies the following condition.

  • Value group of $K^\text{perf}$ (perfect closure of $K$) is $\bigcup_{n\geqq1}(1/p^n) \Bbb{Z}$.

$K$ should be like the form $ \Bbb{F}_q((t))$, so I need to determine $q$.

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    $\begingroup$ I think that using Eisenstein polynomials will show that, for every local field, the value group of $K^\text{sep}$ is $\mathbb Q$. $\endgroup$
    – LSpice
    Commented May 23, 2022 at 14:52
  • $\begingroup$ Sorry, it was typo. Perfect closure was is my question. $\endgroup$
    – Duality
    Commented May 23, 2022 at 15:00
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    $\begingroup$ Then the question is trivial; take $q = p$. $\endgroup$
    – LSpice
    Commented May 23, 2022 at 15:36

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