The automorphism group of a function field

Abstract:

Let $k$ be an algebraically closed field, $K$ a function field in one variable over $k$ , and $G$ a nontrivial finite group. It is proven that there exist infinitely many Galois extensions $L/K$ such that $\operatorname {Gal} (L/K)$ is isomorphic to $G$ , and $\operatorname {Gal} (L/K) = {\operatorname {Aut} _k}(L)$. This extends to arbitrary characteristic, a result first proven in the case $k = \mathbb {C}$ by Greenberg in 1974.