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 $K\mathrm{/}F$ with $\mathrm{A}\mathrm{u}\mathrm{t}(K\mathrm{/}F)\cong G$ that is ramified over exactly $r$ places of $F$? We conjecture that the answer is $\le 1$ 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 $\le 4[F:\mathbb{Q}]$. We show that assuming Schinzel’s Hypothesis H the answer is always $\le 1$ if $F$ is a number field. We show unconditionally that the answer is always $\le 1$ 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 $K\mathrm{/}F$ (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 $N_{S_n}(H)\mathrm{/}H\cong G$.