On Galois theories

The classical Galois theory of fields and the classification of covering spaces of a path-connected, locally path-connected, and semi-locally simply connected space (which will be referred to as the Galois theory of covering spaces) appear very similar. We study the connection of these two Galois theories by generalizing them in categorical language as equivalences of certain categories. This is commonly known as Grothendieck’s formulation of Galois theory. These equivalences of categories can then be related to each other by considering covers of Riemann surfaces, providing a link between the Galois theory of fields and the Galois theory of covering spaces. In particular, we find a link between the Galois group in field theory and the fundamental group in topology. We contextualize this link by considering a topological proof of the Abel-Ruffini theorem (the insolvability of quintics).