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 $k[t]$ ?

completenonarchimedean field). – George Lowther Aug 23 '12 at 11:07