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A. Adrian Albert, "Fundamental Concepts of Higher Algebra"

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 fl;lndamental

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 theor¥.

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.

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