What are all etale morphisms $X \to \operatorname{Spec}\mathbb Z$?
I know the characterization over $\operatorname{Spec}(k)$ where $k$ is a field. These are disjoint union of finite separable field extensions.
So to answer my actual question, I have that every base change of $X$ to $\operatorname{Spec} \mathbb Z_p$ is the disjoint union of finite separable field extensions.
How do I know what the actual $X$ is?
Best Answer
In general, weird morphisms can occur but if you restrict to connected finite étale covers $ X \longrightarrow \operatorname{Spec}(\mathbb{Z})$ then there is only the identity. In a more fancy way, the étale fundamental group of $\operatorname{Spec}(\mathbb{Z})$ is trivial. If you drop out the connectedness then you get some copies of the identity. Its proof uses what you mentioned, first make some reduction modulo $p$ primes and then use the Hermite-Minkowski theorem to compute certain discriminants of number fields.