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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 equicaracteristic, if $k=\overline {\mathbb F_p}$, is there any finite extension of $k[[t]]$ which does not arise from a finite extension of $k[t]$ ?

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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. – S. Carnahan Aug 23 '12 at 8:40
I believe the result Scott is referring to is "Krasner's Lemma". – Jason Starr Aug 23 '12 at 10:40
The answer is no, for the reason S. Carnahan gave (think he meant complete nonarchimedean field). – George Lowther Aug 23 '12 at 11:07
@S. Carnahan: Thank you very much for this answer. Is there any way to have the same result on a purely inseparable extension of $k[[t]]$ ? – Cyrille Corpet Aug 23 '12 at 11:25
I think you mean $k((t))$ (and not $k[[t]]$) and $k(t)$ (and not $k[t]$). – Damian Rössler Aug 23 '12 at 12:19

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