Let $R$ be the ring of integers of some $p$-adic field $K$ (finite over $\mathbb{Q}\_p$) with uniformizer $\pi$ and residue field $k$. I'd like to understand the finite etale extensions of $R((x)) := R[[x]][x^{-1}]$.


Note that $R((x))$ is a non-local PID with uncountably many prime ideals, all of which are generated by elements of the form $\pi^n + r\_1x + r\_2x^2 + \cdots$


One special prime is just the prime $\pi$, at which the residue field is $k((x))$.


**Question:** Is every connected finite etale extension of $R((x))$ given by $R'((x^{1/e}))$ where $R'$ is the ring of integers in some finite unramified extension $K'$ of $K$ and $e$ is coprime to $p$? If not, is it at least dominated by such an extension?


I'd also appreciate any relevant references.



what are the finite etale covers of $\mathbb{Z}_p((x))$?
The answer is affirmative in the "dominated by" aspect, with $R$ any complete discretely valued field whose fraction field $K$ has characteristic 0 and residue field $k$ has characteristic $p \ge 0$ (using $e$ not divisible by $p$; i.e., $e \in k^{\times}$). One cannot do better since the absence of various roots of unity from $K$ can mess up the "easy" Galois theory description of intermediate extensions. This is a standard application of Abhyankar's Lemma, as follows.


In the ring $A = R[\![x]\!]$, the height-1 prime $P = (x)$ has residue field $K$ of characteristic 0 (and uniformizer $x$), so all ramification at $P$ in any finite extension $L'$ of $L := {\rm{Frac}}(A) = {\rm{Frac}}(R(\!(x)\!))$ is *tame*. Hence, if $A'$ is the (module-finite) normalization of $A$ in $L'/L$ and $A'[1/x]$ is etale over $A[1/x] = R(\!(x)\!)$ then there is only tame ramification over ${\rm{Spec}}(A)$. It follows by Abhyankar's Lemma (as proved in an appendix of the book of Freitag-Kiehl on etale cohomology, for example) that ${A'}^{\rm{sh}} = A^{\rm{sh}}[x^{1/e}]$ for an integer $e \ge 1$ with $e \in k^{\times}$. Thus, $A' \subset A^{\rm{sh}}[x^{1/e}]$. 


Since $A$ is complete local noetherian, so $A^{\rm{sh}}$ is exhausted by local finite etale extensions of $A$, which in turn correspond to finite separable extensions of $k$, as we vary through the local finite etale extensions $R'$ of $R$ the resulting finite etale $A$-algebras $R'[\![x]\!]$ exhaust $A^{\rm{sh}}$. Hence, for big enough such $R'$ we see that $A' \subset R'[\![x]\!][x^{1/e}]$. 


The extension of Dedekind domains 
$$R(\!(x)\!) = A[1/x] \rightarrow R'(\!(x^{1/e})\!)$$
is clearly finite etale, so any intermediate module-finite Dedekind domain is also finite etale over $A[1/x]$. Thus, the finite etale $R(\!(x)\!)$-algebra domains are precisely the module-finite subalgebras of such extensions $R'(\!(x^{1/e})\!)$ for $R'$ local finite etale over $R$ and $e \ge 1$ not divisible by ${\rm{char}}(k)$. That is the "dominated by" assertion we wanted to prove.


