Explicit Galois theory in positive characteristic

Abstract

In Chapter 1, we will go over some preliminaries leading up to defining the Galois group. This leads us to two definitions of the Galois group in Chapter 2, one original definition and a more modern definition. With this definition, we will prove tools to compute the Galois group of polynomials of degree 3 and 4. In Chapter 2, we will do this for polynomials over fields of characteristic not equal to 2, but in Chapter 3 we give the tools to compute Galois groups over fields of characteristic 2 as well. Finally in Chapter 4, we will introduce some more important theorems that will allow us to prove the van der Waerdens Theorem, stating that for a random polynomial over $\Q$ the probability of the Galois group being $S_n$ is 1. Furthermore, we use these theorems to look at two different families of polynomials, for polynomials of the form $x^n+ax^k+b$ we can determine the Galois group to be $S_n$ for all degrees $n$ under a few conditions. For the other family, additive polynomials, we will show that the Galois group is a subgroup of $GL_n$.