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Is there any finite extension of $\mathbb Q_p$ which is not the completion of a finite extension of $\mathbb Q$ at some place over $p$?

Analogously in equicharacteristic, if $k=\overline {\mathbb F_p}$ , is there any finite extension of $k((t))$ which doesn't arise from a finite extension of $k(t)$?

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    $\begingroup$ If I'm not mistaken, the isomorphism type of a simple extension of a nonarchimedean field is locally constant with respect to the coefficients of a defining polynomial, so at least in characteristic zero, you can choose suitably nearby rational coefficients to get a number field that completes to the $p$-adic field you want. $\endgroup$
    – S. Carnahan ♦
    Commented Aug 23, 2012 at 8:40
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    $\begingroup$ I believe the result Scott is referring to is "Krasner's Lemma". $\endgroup$
    – Jason Starr
    Commented Aug 23, 2012 at 10:40
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    $\begingroup$ The answer is no, for the reason S. Carnahan gave (think he meant complete nonarchimedean field). $\endgroup$
    – George Lowther
    Commented Aug 23, 2012 at 11:07
  • $\begingroup$ @S. Carnahan: Thank you very much for this answer. Is there any way to have the same result on a purely inseparable extension of $k[[t]]$ ? $\endgroup$
    – Cyrille Corpet
    Commented Aug 23, 2012 at 11:25
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    $\begingroup$ I think you mean $k((t))$ (and not $k[[t]]$) and $k(t)$ (and not $k[t]$). $\endgroup$
    – Damian Rössler
    Commented Aug 23, 2012 at 12:19

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