Finite etale maps to the line minus the origin I am trying to determine the etale fundamental group of $V = A^1 - \{0\}$ over an algebraically closed field $k$. I am trying to stay in the comfortable zone of non-singular varieties.
To do this, I wonder if there is an easy way to determine all finite etale maps $f:W\to V$ where $W$ is a non-singular variety over $k$.
Any hints how to find all these maps? can I compute the etale fundamental group without finding these?
By finiteness, I guess $W$ must be a finite union of space curves and points.
As a start, I checked what are the finite etale automorphisms of $V$, these are simply given by $a \mapsto a^n$.
Thanks!
 A: I'm not sure how elementary/algebraic you will consider this. Let $k$ be an algebraically closed field of characteristic $0$.
Let $W \to V$ be an etale map. Assume $W$ is connected; a general etale map will then be the disjoint union of several examples of this sort. Since $W$ is etale over $V$, we know that $W$ is smooth and one dimensional. Let $\overline{W}$ be the complete curve containing $W$, so we have a map $\overline{W} \to \mathbb{P}^1$. Let the degree of this map be $n$; let $g$ be the genus of $\overline{W}$; let $e^0_1$, ..., $e^0_r$ be the ramification degrees of the points over $0$ and let $e^{\infty}_1$, ..., $e^{\infty}_s$ be the ramification degrees of the points over $\infty$.
The Riemann-Hurwitz formula gives
$$2g-2 = -2n + \sum (e^0_i-1) + \sum (e^{\infty}_i-1).$$
(This is the step which is invalid in positive characteristic.)
The right hand side is
$$-2n + \sum e^0_i + \sum e^{\infty}_i - r -s = -2n+n+n-r-s=-r-s.$$
So
$$2g+r+s=2.$$
But $g \geq 0$ and $r$ and $s \geq 1$. So this can only hold if $g=0$ and $r=s=1$. The fact that $g=0$ means that $\overline{W} \cong \mathbb{P}^1_k$. The fact that $r=s=1$ means that there is one point of $\overline{W}$ lying over $0$, and one point lying over $\infty$; without loss of generality, let those points be $0$ and $\infty$.
So our map is of the form $t \mapsto p(t)/q(t)$ for some relatively prime polynomials $p$ and $q$, and the preimages of $0$ and $\infty$ are $0$ and $\infty$. So the only root of $p$ can be $0$, and $q$ can have no roots at all. We conclude that our map is of the form $t \mapsto a t^n$, as desired.

You definitely can give purely algebraic proofs that every curve embeds in a complete curve, and of Riemann-Hurwitz. I feel like one should be able to give pretty elementary ones, but I don't know a reference which does it in an elementary way. 
A: Suppose $X$ is a scheme locally of finite type over $\mathbb C$. Then the category of finite étale covers of $X$ is equivalent to the catgory of finite analytic étale covers over $X^{an}$, where $X^{an}$ is the analytic space canonically associated to $X$. The equivalence associates to the étale cover  $X'\to X$ its analytification $(X')^{an}\to X^{an}$. 
In your case this implies that the only étale covers of $\mathbb G_m=V=\mathbb A_k^1 \setminus \{0\}$ are the morphisms you mentioned $\mathbb G_m\to \mathbb G_m:z\mapsto z^n$. 
The same result is true over any algebraically closed field of characteristic $0$, and implies that the algebraic fundamental group of $ \mathbb G_m $ is the profinite completion $\pi_1^{alg}(\mathbb G_m )=\hat{\mathbb Z}$ of  the topological fundamental group  $\pi_1^{top}(\mathbb G_m^{an})=\mathbb Z $ . I recommend extreme prudence in characteristic $p$, since as far as I know even the structure  of the algebraic fundamental group of $\mathbb A^1_k$ is not known in characteristic $p$ !
Bibliography The equivalence of categories  mentioned above is due to Grauert-Remmert. There is a shorter proof in Grothendieck's SGA 1, Théorème 5.1, which however uses Hironaka's resolution of singularities in characteristic zero.
