Course abstract

In this introductory course on Galois theory, we will first review basic concepts from rings and fields, such as polynomial rings, field extensions and splitting fields. We will then learn about normal and separable extensions before defining Galois extensions. We will see a lot of examples and constructions of Galois groups and Galois extensions. We will then prove the fundamental theorem of Galois theory which gives a correspondence between subgroups of the Galois group and intermediate fields of a Galois extension. We will then cover some important applications of Galois theory, such as insolvability of quintics, Kummer extensions, cyclotomic extensions.