Infinite Galois theory and profinite groups

Abstract

This paper aims to provide an exposition on infinite Galois extensions and profinite groups as well as some applications of Galois theory through six chapters. We will begin with an introduction to topological groups, inverse system and inverse limits of topological spaces and topological groups. Thereafter, we will introduce profinite groups and provide a characterisation of profinite groups as Galois groups by proving Krull’s Theorem on infinite Galois extensions. To end it off, we will explore the norm and trace maps of Galois extension and their applications including the Hilbert’s Theorem 90 and its generalisation in the study of group cohomology.