Let $R$ be a DVR with maximal ideal $xR$, and assume that $R$ is not complete in the $xR$adic topology. Let $\hat{R}$ be the completion of $R$ in the $xR$adic topology. Set $K=Q(R)$, the fraction field of $R$, and set $K'=Q(\hat{R})$. Must the degree of the field extension $K'/K$ be infinite? The answer is "yes" for all the examples that I know, and it feels like it should be true in general, but I don't see it in general.

Nagata's example (E3.3) shows that you can have $[K':K]=p<\infty$. Thanks to Bruce Olberding for the pointer. 

