TITLE: Etale covers of the affine line
QUESTION [26 upvotes]: In characteristic p there are nontrivial etale covers of the affine line, such as those obtained by adjoining solutions to x^2 + x + f(t) = 0 for f(t) in k[t].  Using an etale cohomology computation with the Artin-Schreier sequence I believe you can show that, at least, the abelianization of the absolute Galois group is terrible.
What is known about the absolute Galois group of the affine line in characteristic p?  In addition, can spaces which are not A^1 (or extensions of A^1 to a larger base field) occur as covers?

REPLY [10 votes]: As stated by David Speyer and Clark Barwick, the awnser to the second question is the following:
Any smooth projective curve $C$ defined over a field $k$ of positive characteristic $p$ can be realized as a finite cover of the projective line only ramified above one point.
Here is a short constructive proof only based on Riemann-Roch theorem. It can be considered as an illustration of Kedlaya's proof, only dealing with curves.

First of all, there exists a generically étale finite cover $C\to\mathbf P^1$, induced by a rational function $f\in k(C)$ (in fact, any element of $k(C)-k(C)^p$ will do the job).
Denote by $R\in$ Div$(C)$ the (reduced) ramification divisor of the above cover (i.e. the ramified points are couted without multiplicity). From Riemann-Roch theorem, for large $n$, there exist a rational function $g\in k(C)$ having a pole of order $n$ at each point of the support of $R$. We may take $n$ strictly greater than $\frac{\deg(f)}p$.
Then, the rational function $h=f+g^p$ induces a cover $C\to\mathbf P^1$ only ramified above infinity.