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Let $R=\mathbb{Z}_{(p)}$. What are finite etale algebras over $R$? Let $R \to A$ be a finite etale map where $Spec A$ is connected. Then $A$ is the normalisation of $R$ in $Frac(A)$. Therefore, they all look like $\mathcal{O}_{K}$ for some number field $K$. However, there are restrictions. Namely, the extension $K/\mathbb{Q}$ should be unramified at $p$. Thus, I think that $\pi_{1}^{et} (R)$ is the Galois group of the maximal unramified at $p$ extension of $\mathbb{Q}$. However, I haven't found any mentions of this object in google so maybe I'm wrong. Am I wrong? If not, is it related to the maximal unramified extension of $\mathbb{Q}_p$ in some sense? In particular, is $\pi_{1}^{et} (R) = \hat{\mathbb{Z}}?$

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  • $\begingroup$ The answer to the questions in the last two sentences is 'no'; the unramified-at-p global Galois group is huge (not even topologically finitely generated), so it is much bigger than Z-hat. $\endgroup$ 2 days ago
  • $\begingroup$ @DavidLoeffler Thanks for covering all the bases -- I honestly must have glossed over the last two sentences. $\endgroup$ yesterday

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Yes: $\pi_1^{\mathrm{et}}(\mathrm{Spec}(\mathbb{Z}_{(p)})$ is $\mathrm{Gal}(K/\mathbb{Q})$ where $K$ is the maximal extension of $\mathbb{Q}$ unramified at $p$. This is not special to $\mathbb{Z}_{(p)}$ and holds for any normal domain (cf. Tag 0BQM)

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