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For a Dedekind domain $A$ we have the following result relating torsion elements of the class group to (mostly) unramified extensions:

If $a\in A$ is such that there exists an ideal $I$ with $I^n=(a)$, then for $B$ the integral closure of $A$ in $Quot(A)(a^{1/n})$, we have $B/A$ unramified for all prime ideals not containing $(n)$.

In the number field case, ideals containing $(n)$ are nontrivial, but in the riemann surface case, or the function field case for $p$ coprime to $n$ this is an everywhere unramified cover.

My intuition is that everywhere unramified extensions correspond to covering spaces, and for $a$ not a unit, we should get a real geometric cover, rather than a base field extension. So the question is then, what does this cover look like, and whats really going on here?

The field extension is also quite close to a kummer extension, so in some cases, the extension will be abelian galois and unramified, which are words that I have read together in the context of class field theory. I dont know any class field theory however, so it may just be a coincidence.

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