Automorphism Groups of Finite Extensions of Fields and the Minimal Ramification Problem

We study the following question: given a global field F and finite group G, what is the minimal r such that there exists a finite extension K/F with Aut (K/F) G that is ramified over exactly r places of F? We conjecture that the answer is 1 for any global field F and finite group G. In the case when F is a number field we show that the answer is always 4F: Q. We show that assuming Schinzel's Hypothesis H the answer is always 1 if F is a number field. We show unconditionally that the answer is always 1 if F is a global function field. We also show that for a broader class of fields F than previously known, every finite group G can be realized as the automorphism group of a finite extension K/F (without restriction on the ramification). An important new tool used in this work is a recent result of the author and C. Tsang, which says that for any finite group G there exists a natural number n and a subgroup H Sₙ of the symmetric group such that Nₒ䂸 (H) /H G.