Title:

On Galois groups of Salem polynomials

The primary topic of this thesis is the Galois group of an irreducible Salem polynomial and how the precise structure of the roots of such a polynomial is reflected in its Galois group. The main goal is the proof of the structure of these Galois groups always being a Semidirect product N x G/N. This is a stronger result than is known for reciprocal polynomials in general: [37]. Here N will be a certain normal subgroup of the Galois group, G, of our irreducible Salem polynomial f(x), and G/IN ~ Galois group of the Trace polynomial g(y), for f(x) = xdg(x+ 1/x) and f(x) being of degree 2d. We give here an overview of our work. • In Chapter 1 we give some basic prerequisites for our work and we conclude with the result stated in [19] about the Galois group of a Salem polynomial. • In Chapter 2 we give an introduction on what will follow on the structure of an irreducible Salem polynomial of degree six. • Chapter 3 plays two roles. On the one hand it includes material concerning the structure of the Galois group of an irreducible Salem polynomial of minimum order. On the other hand it demonstrates how the structure of the roots of a Salem polynomial can be used to exclude certain groups from being Galois groups of these polynomials. Here we exclude the Dihedral group. • In Chapter 4 we answer the question of when the smallest order Galois group can occur and what the structure of the Galois field will be in that case. • In Chapter 5 we give some background on group extensions. We conclude with demonstrating how in some cases the Galois group is a Semidirect product and what is preventing us from extending this to every Galois group of an irreducible Salem polynomial. • In Chapter 6 we prove that the Galois group of a degree six irreducible Salem polynomial will always be the Semidirect product stated above. • In Chapter 7 we analyse the Discriminant of these polynomials.
