Realising $S_n$ and $A_n$ as Galois groups Over $\mathbb{Q}$ : an introduction to the inverse Galois problem

Abstract

Given a field $k$ and a finite group $G$, is there a Galois field extension $K|k$ such that its Galois group is isomorphic to $G$? Such an innocent question and yet it remains unsolved: this is what is known as the Inverse Galois Problem. In the present Bachelor thesis we show that this question has a positive answer if the field is $\mathbb{Q}$ and the group is either $S_n$ or $A_n$, following the strategy devised by David Hilbert in his paper Über die Irreduzibilität ganzer rationaler Funktionen mit ganzzahligen Koeffizienten (1892). We start with two basic examples and an exposition of relevant results from algebraic number theory, and then move on to proving Hilbert’s Irreducibility Theorem. As a consequence, we prove that the symmetric group $S_n$ and the alternating group $A_n$ are realisable as Galois groups over the field of rational numbers $\mathbb{Q}$.