# tamely branched cover over P^1

k is an algebraically closed field, X is a smooth, connected, projective curve over k. f: X-->P^1 is a finite morphism. Let t be a parameter of P^1, suppose f is etale outside t=0 and t=\infty, and tamely ramified over these two points. Prove that f is a cyclic cover, i.e., K(X)=k(t)[h]/(h^n-ut), u is a unit in field k.

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By Hurewicz's formula I can prove the ramification indices at these two points are n, n is the degree of f. But I can't see why it must be a cyclic cover. –  Taisong Jing Dec 26 '09 at 21:00
By the way, I think you mean (Adolf) Hurwitz, rather than (Wittold) Hurewicz. –  Pete L. Clark Dec 26 '09 at 23:38
yes, you are right~ –  Taisong Jing Dec 27 '09 at 4:25

Then observe that the tame fundamental group of $\mathbb{P}^1$ minus two points is procyclic. This follows from a comparison theorem of Grothendieck, which allows you to reduce to the case $k = \mathbb{C}$.