On the realization of subgroups of $PGL(2,F)$, and their automorphism groups, as Galois groups over function fields

TL;DR

This paper proves that any finite subgroup of PGL(2,F) can be realized as a Galois group over the rational function field F(x), and develops a descent theory to realize automorphism groups as Galois groups.

Contribution

It provides an elementary proof for realizing finite subgroups of PGL(2,F) as Galois groups over F(x) and introduces a descent theory for automorphism groups.

Findings

Finite subgroups of PGL(2,F) are realizable as Galois groups over F(x)

Develops a descent theory for automorphism groups of these subgroups

Enables realization of automorphism groups as Galois groups

Abstract

Let $F$ be any field. We give a short and elementary proof that any finite subgroup $G$ of $PGL(2,F)$ occurs as a Galois group over the function field $F(x)$. We also develop a theory of descent to subfields of $F$. This enables us to realize the automorphism groups of finite subgroups of $PGL(2,F)$ as Galois groups.