Preface The existence of the digital computer and other devices using binary digits has resulted in a renewal of interest in the mathematical theory of finite fields. The main exposition of the foundations of the subject was written over fifty years ago and has been out of print for many years. A new exposition using the modern theory of algebraic extensions of arbitrary fields seems very timely and is the principal reason for the preparation of the present text. We give here a compact and self-contained exposition of the fundamental concepts of modern algebra which are needed for a clear understanding of the place of finite field theory in modern mathematics. In the first chapter we introduce the primitive concepts on which algebra is based and go on to a fairly complete exposition of the basic notions about finite groups, including the theory of composition series in general form. In chapter ii we present the concepts of ring, ideal, difference ring, Mid, polynomials over a ring, and the simple cases of the integral domain of ordinary integers and of polynomials in one indeterminate over a field. The material of this chapter is then the foundation for the later theory of algebraic extensions. A logical exposition of the theory of field extension is impossible without a preliminary exposition of the theory of vector spaces, linear mappings, and matrices. This is then the subject of chapter iii. All of the theory of matrices which seems to be appropriate is presented, including the theory of matrix equivalence over a polynomial ring and the theory of similarity. Since the theory of quadratic forms and orthogonal equivalence seems out of context, it is not presented. In chapter iv we provide a substantial exposition of the theory of alaebraic extensions of fields, including the Artin version of the galois theory. The chapter ends with a new and remarkably simple proof of the normal basis theorem for cyclic fields. The real reason for the book is chapter v, in which we present a modern and improved exposition of the foundations of the theory of finite fields. Galois theory and group theory are used where they belong, and many proofs are improved thereby. It is hoped that this new text will prove to be a valuable aid not only to those who wish to learn the basic theorems on finite fields but to those who wish to refresh their knowledge of modern algebra or to review a series of courses on modern algebra which they may have just completed. The text should also be usable as a first course in the subject, provided that it is recognized that the presentation is exceedingly compact and requires slow and careful classroom discussion. The preparation of this text was supported by the Department of Defense. The author also wishes to acknowledge the very kind assistance of Drs. W. A. Blankinship and E. C. Paige, who proofread the original manuscript with great care.