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 ≤ 4 [ F : 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 ⩽ S n of the symmetric group such that N S n ( H ) / H ≅ G .