Galois Extensions

Abstract

In this chapter, we arrive at the main theorems of Galois theory. Following the material covered in previous chapters, we are now equipped to define Galois extensions, i.e. the extensions which are both normal and separable. One of the most central results is Galois’ correspondence between the subgroups of the Galois group of such an extension and the intermediate subfields of it. In the exercises, we find many examples and interesting properties of Galois extensions. Several exercises are related to the inverse problem in Galois theory: to construct a Galois extension of the rational numbers with given group as its Galois group. In a series of exercises, this problem is solved for different groups of small orders and for all cyclic groups using a special case of Dirichlet’s theorem on primes in arithmetic progression, which is proved in the next chapter.