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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.

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    What about $\mathbb{Z}_p$ and $\mathbb{F}_p[[t]]$? – Alex Youcis Jul 29 '19 at 10:01
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    @AlexYoucis does the fraction field of $\mathbb{F}_p[[t]]$ have characteristic 0? –  Jul 29 '19 at 10:08
  • We don't even need the residue field, I think. The fraction field $K:=\operatorname{Frac} R_1$ has a unique valuation $v:K \to \mathbb{Z} \cup {\infty}$ such that $R_1={x \in K ; | ; v(x) \ge 0 }$. – Pol van Hoften Jul 29 '19 at 11:25
  • @user45878 but the valuation is not guaranteed to satisfy the property "such that...". –  Jul 29 '19 at 11:42
  • @ParthivBasu why is there only one complete DVR in mixed characteristic? –  Jul 29 '19 at 12:12
  • @ParthivBasu I think the ring of integers in $\mathbb{Q}_2[x]/(x^2+2)$ is a complete DVR with residue field $\mathbb{F}_2$ that is not isomorphic to $\mathbb{Z}_2$. I could be wrong. –  Jul 29 '19 at 12:26
  • Have you looked into ramified Witt rings? They allow to reconstruct the valuation ring from a uniformizer and the residue field. This might give a positive answer to the uniqueness. – Claudius Jul 29 '19 at 13:21
  • @misa I am pretty sure this is true in the case of discretely valued fields, do you have a counterexample in mind? – Pol van Hoften Jul 29 '19 at 15:41

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