Finite extensions of number fields as coverings In chapter II on completions in Serge Lang's Algebraic Number Theory, he says that it is useful to think of finite extensions of a number field as coverings. I am confused about it. I thought maybe he means there is some kind of surjective homomorphism from the extension of a number field onto it. However, field homomorphisms are injective so it is impossible for a nontrivial extension of a number field to surject onto it.
Does anyone have ideas what he means by coverings here and how is that useful? Thank you.
 A: There's the theory of finite étale covering of a scheme (part of Grothendieck's Galois theory). In the case of $\mathrm{Spec}(K)$, the connected finite étale coverings are of the form $\mathrm{Spec}(L) \to \mathrm{Spec}(K)$, where $L/K$ is a finite separable field extension.
Grothendieck's Galois theory generalises classical Galois theory, but can also be applied to a topological space in which case one recovers (at least for finite degree) the relation between coverings and subgroups of the fundamental group.
The keywords to search for are Grothendieck's Galois theory, étale morphisms and the étale fundamental group and Galois categories.
Even without these relatively high-level concepts that make the analogies precise, it can help to consider the formal similarities between Galois theory and covering space theory. In both cases there's a group, the subgroups of which control the whole covering theory/theory of separable field extensions. In both cases, there's a notion of normal coverings and normal extensions which correspond to normal subgroups. These analogies can be continued and it would be a worthwhile exercise to try and find more analogies between Galois theory and covering spaces.
The geometric picture can be refined if one considers ring of integers. $\mathrm{Spec}(\mathcal O_L) \to \mathrm {Spec}(\mathcal O_K)$ is a finite morphism of schemes, but not necessarily étale because there can be ramification. So in some sense, we have a ramified cover, with very similar properties to, say a nonconstant holomorphic map between compact Riemann surfaces. In the geometric picture, thinking about $\mathrm{Spec}(\mathcal O_K) $ as a curve, for example taking the integral closure of a non-integrally closed subring is like desigularization of a singular algebraic curve. You can find some material with pictures on this in Neukirch section I.13 "one-dimensional schemes".
