Vector Spaces and Field Extensions

Abstract

We begin this chapter with some basic facts about vector spaces. These will be familiar (at least in the case of real vector spaces) to those readers who have studied linear algebra. We then focus our attention on the particular case of a field extension. A number of properties of field extensions are discussed. Let F be a field and \(f(x)\in F[x]\) a nonconstant polynomial. We demonstrate how to create a field extension in which f(x) splits into a product of polynomials of degree 1. This leads to a classification of all finite fields.