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.

• 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.
– TJCM
Dec 26 '09 at 21:00
• By the way, I think you mean (Adolf) Hurwitz, rather than (Wittold) Hurewicz. Dec 26 '09 at 23:38
• yes, you are right~
– TJCM
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 $k = \mathbb{C}$.
• For those who like references, the comparison theorem which gives the result about the tame fundamental group of $\mathbf P^1$ minus two points as a corollary is SGA 1 Exp. XIII 2.12.