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The purpose of this chapter is to review some basic facts that will be needed in the book. The discussion is not intended to be complete, nor are all proofs supplied. We suggest that the reader quickly skim this chapter (or skip it altogether) and use it as a reference if needed.

In this chapter, we discuss properties of polynomials that will be needed in the sequel. Since we assume that the reader is familiar with the basic properties of polynomials, some of the present material may constitute a review.

In this chapter, we will describe several types of field extensions and study their basic properties.

Let us recall a few facts about separable polynomials from Chapter 1.

In this chapter, we discuss the structure of an arbitrary field extension < . We will see that for any extension < . there exists an intermediate field < () < whose upper step () < is algebraic and whose lower step < () is , that is, there is no nontrivial polynomial dependency (over ) among the elements of , and so these elements act as “independent variables” over . Thus, () is the field of all rational functions in these variables.

Galois theory sits atop a structure of work began about 4000 years ago on the question of how to solve polynomial equations algebraically , that is, how to solve equations of the form by applying the four basic arithmetical operations (addition, subtraction, multiplication and division), and the taking of roots, to the coefficients of the equation and to other “known” quantities (such as elements of the base field).

The traditional Galois correspondence between intermediate fields of an extension and subgroups of the Galois group is one of the main themes of this book. We choose to approach this theme through a more general concept, however.

In this chapter, we pass from the highly theoretical material of the previous chapter to the somewhat more concrete, where we apply the results of the previous chapter to some special Galois correspondences.

In this chapter, we take a closer look at a finite extension < from the point of view that is a vector space over . It is clear, for instance, that any ∈ () is a linear operator on over . However, there are many linear operators that are not field automorphisms. One of the most important is multiplication by a fixed element of , which we study next.

In this chapter and the next, we study finite fields, which play an important role in the applications of field theory, especially to coding theory, cryptology and combinatorics. For a thorough treatment of finite fields, the reader should consult the book , by Lidl and Niederreiter, Cambridge University Press, 1986.