ABSTRACT
In this chapter we make a big connection between field theory and the theory of groups called Galois theory. Such connections are ways in which powerful tools can be created for proving difficult mathematical results. The result which motivated Evariste Galois to develop this theory and its connections is the investigation of solvability by radicals. For the reader who is interested in the history of mathematics, Galois' biography is quite interesting and dramatic.
In Section 10.1 we relate fields and groups via field homomorphisms of an extension field which fix the base field, thus arriving at Galois groups. We need Section 10.2 so that we can compute these Galois groups in a practical way. Section 10.3 is a bit off the beaten path, but at the very least the reader should be aware of the important results arrived at in this section. In Section 10.4 we introduce the notion of a splitting field and prove two important results linking the Galois group and the corresponding field extension. In Section 10.5 we introduce separable degree and link this concept to the size of the corresponding Galois group. In Section 10.6 introduce the notion of a Galois extension as we compare the lattice of subgroups of the Galois group and the corresponding lattice of intermediate fields in the corresponding field extension. In Section 10.7 we give the promised proof of Theorem 10.7 presented in Section 10.6 as well as prove a foundational result of Artin. In Section 10.8 we summarize our investigation in this chapter and prove the Fundamental Theorem of Galois Theory. In Section 10.9, we review Chapter 6 emphasizing the important concepts and results given therein regarding solvable groups. Finally, in Section 10.10, we investigate the notion of solvability by radicals.
