ABSTRACT
In this chapter we set the stage for the next chapter on Galois theory. In Section 9.1, we remind the reader of some definitions, concepts and results as well as introduce the definition of algebraic. In Section 9.2, we investigate field extensions which allows us to look at chains of fields, an important idea in Galois theory. This also allows us to investigate some famous geometric impossibilities in Section 9.3. In Section 9.4, we look at some particular and important field extensions, and in Section 9.5, we prove the existence of some of these structures we introduced. Finally, in Section 9.6, we completely classify finite fields.
