Alexandros Halivopoulos: Galois Groups and Fundamental Groups
Master thesis
Tid: Ti 2025-02-04 kl 11.00 - 12.00
Plats: Kovalevskyrummet
Respondent: Alexandros Halivopoulos
Abstract.
Ever since Galois theory emerged, there has been an open problem called The inverse Galois problem. It states whether or not any finite group \(G\) can occur as the Galois group of a finite extension over a fixed base field \(k\). This depends on the base field \(k\). For \(k = \mathbb{C}(t)\) the problem has a positive answer and a proof will be provided with the help of Riemann surfaces. When \(k = \mathbb{Q}\) the problem remains open.
In this project, a promising approach will be presented concerning the algebraic fundamental group, which will allows us to turn a finite group \(G\) with trivial center and a rigid system of rational conjugacy classes into a Galois group over \(\mathbb{Q}\). To obtain the algebraic fundamental group we will present the theory in the following way: In the first chapter the Galois theory for both finite and infinite extensions will be presented and also Groethendieck‘s reformulation of the main Galois theorem will be stated and proved. The latter serves as an abstraction and will allow us to draw analogies between the two central objects of the project, namely the Galois groups and the Fundamental groups. In the second chapter, the focus will be on covers and the fundamental group of a space and the analogy between them with field extensions and the absolute Galois group. The third chapter is about Riemann surfaces for which we will obtain a link between the Galois theory of fields and that of covers. It will be proven that finite etale algebras over the field of meromorphic functions of a fixed Riemann surface corresponds up to isomorphism to finite branched covers of Riemann surfaces. The last chapter will be dedicated to obtaining the algebraic fundamental group via the theory of algebraic curves by relating it to the theory presented in the third chapter.