Computation of Galois groups and corresponding polynomials

Abstract

In this thesis we will look into the process of calculating Galois groups and computing polynomials that correspond to a certain Galois group. In the first chapter we will state and prove some useful theorems - building a theoretical foundation. In the second chapter we will investigate and improve an algorithm described by Cohen. This will enable us to find the Galois group of all polynomials with integer coefficients. In the last chapter we will look at transitive subgroups of S3 and S4 and state infinitely many polynomials corresponding to each of the subgroups. We will then use elementary symmetric functions to find out more about the group structure of subgroups of S4 and their corresponding subfields.