Galois extensions ramified only at one prime

TL;DR

This paper investigates restrictions on finite groups as Galois groups of extensions over rational and function fields, focusing on ramification at only one prime, and extends known results to new classes of groups.

Contribution

It strengthens existing results on dihedral Galois extensions over $\\Q$ and provides new restrictions for symmetric and dihedral groups over $\\F_q(t)$.

Findings

Restrictions on dihedral groups over $\\Q$

Restrictions on symmetric groups over $\\F_q(t)$

Non-solvable groups are ruled out in certain cases

Abstract

This paper gives some restrictions on finite groups that can occur as Galois groups of extensions over $\Q$ and over $\F_q(t)$ ramified only at one finite prime. Over $\Q$, we strengthen results of Jensen and Yui about dihedral extensions and rule out some non-solvable groups. Over $\F_q(t)$ restrictions are given for symmetric groups and dihedral groups to occur as tamely ramified extension over $\F_q(t)$ ramified only at one prime.