Topics in Inverse Galois Problem

Abstract

The Inverse Galois Problem asks whether every finite group occurs as the Galois group of some field extension of the rational numbers, Q. This is still an unsolved problem. This project explores various methods to try to answer this question, most notable among which is the rigidity method. In this project, the required theory is presented to arrive at the Rigidity Criterion which is then applied to various finite simple groups, including the Mathieu groups, linear groups and the Monster group, to show their occurrence as Galois groups over Q.