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

TL;DR

This paper investigates the minimal ramification problem for automorphism groups of finite field extensions over global fields, proposing conjectures and providing bounds under various hypotheses, with new constructions for realizing groups as automorphism groups.

Contribution

The paper introduces new bounds on minimal ramification for automorphism groups over global fields and constructs broad classes of fields where any finite group can be realized as an automorphism group.

Findings

Conjecture that minimal ramification is at most 1 for all global fields and finite groups.

Proves bounds of at most 4 times the degree for number fields.

Unconditionally shows minimal ramification is at most 1 for global function fields.

Abstract

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 $\mathrm{Aut}(K/F)\cong G$ 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 $K/F$ (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 $H\leqslant S_n$ of the symmetric group such that $N_{S_n}(H)/H\cong G$.