Recently, I have been trying to prove a theorem about all (non-field) domains $R$ complete and locally compact with respect to an absolute value $|\cdot|$. This begs first the question of classifying all such rings.
In the case that the absolute value is archimedean, we can proceed simply. The metric space completion $\overline{K}$ of the fraction field of $K$ must be isomorphic to $\mathbb{R}$ or $\mathbb{C}$ by the Gelfand-Mazur theorem for Banach algebras. A computation shows that $R$ must be discrete, and there is a classification of discrete subrings of $\mathbb{C}$ saying that either $R=\mathbb{Z}$ or $R=\mathbb{Z}\left[(a+\sqrt{b^2+4a})/2\right]$ where $a$ is a negative integer and $|b|<2\sqrt{-a}$ is also an integer.
The case of non-archimedean absolute values is the one tripping me up. If $R$ were a DVR then life would be simple, since there is a well-known classification of complete DVRs. Can this be shown?
A sufficient answer to this question would also be giving a classification of complete fields, as was done in the archimedean case. It would be then left to me to try to piece together what complete subrings exist.
Thank you in advance for the help.