Subfields of Simple Algebras

In this chapter we set the stage for a systematic study of the Brauer group. The details of the program will be worked out in the next chapter, using two main tools: the cohomology theory that was introduced in Chapter 11, and the properties of subfields of central simple algebras which will be established in the first five sections of this chapter. The last section of the chapter gives appUcations of the theory, including Wedderbum’s Theorem on Finite Division Algebras, and the Cartan-Brauer-Hua Theorem.