Automorphism groups of field extensions and the minimal ramification problem

I will discuss the following problem: given a global field \(F\) and finite group \(G\), what is the minimal \(r\) such that there exists a finite extension \(K/F\) (not necessarily Galois) with \(Aut(K/F) \cong G\) that is ramified over exactly \(r\) places of \(F\)?

Evidence will be given to the conjecture that the answer is \(r=0\) or \(1\) for all \(F, G\). An important new tool used in this work is a recent group-theoretic result which says that for any finite group \(G\) there exists a natural number \(n\) and a subgroup \(H<S_n\) of the symmetric group such that \(N_{S_n}(H)/H \cong G\).

Based on joint work with Cindy Tsang.