# Maximal unramified at p extension of $\mathbb{Q}$

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 $$\pi_{1}^{et} (R) = \hat{\mathbb{Z}}?$$

• 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. 2 days ago
• @DavidLoeffler Thanks for covering all the bases -- I honestly must have glossed over the last two sentences. yesterday

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)