Assume we have two complete DVRs $R_1$, $R_2$ with finite residue fields. Assume their residue fields and fraction fields are isomorphic as abstract fields. Can $R_1$ and $R_2$ be non-isomorphic (as unital rings)?
EDIT: from Serre's textbook on local fields, we know that a complete DVR of equal characteristic with a perfect residue field $k$ is isomorphic to $k[[t]]$.
Given a perfect field $k$, denote by $\mathbb{W}(k)$ the ring of Witt vectors over $k$. For a complete DVR $R$ of unequal characteristic with residue field $k$ and with absolute ramification index $e$ there exists a unique homomorphism $\mathbb{W}(k)\rightarrow R$ commuting with reduction modulo the maximal ideal. This is injective, and $R$ is a free $\mathbb{W}(k)$-module of rank $e$. So we must look for ramified mixed characteristic DVRs.