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 $k[t]$ ?

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 $k(t)$?

Is there any 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 $k[t]$ ?

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 $k[t]$ ?