INTRODUCTION TO ALGEBRAIC THEORIES By A. ADRIAN ALBERT THE UNIVERSITY OF CHICAGO THE UNIVERSITY OF CHICAGO PRESS CHICAGO ILLINOIS COPYRIGHT 1941 BY THE UNIVERSITY OF CHICAGO. ALL RIGHTS RESERVED. PUBLISHED JANUARY 1941. SECOND IMPRESSION APRIL 1942. COMPOSED AND PRINTED BY THE UNIVERSITY OF CHICAGO PRESS, CHICAGO, ILLINOIS, U. S. A. PREFACE During recent years there has been an ever increasing interest in modern algebra not only of students in mathematics but also of those in physics, chemistry, psychology, economics, and statistics. My Modern Higher Alge bra was intended, of course, to serve primarily the first of these groups, and its rather widespread use has assured me of the propriety of both its con tents and its abstract mode of presentation. This assurance has been con firmed by its successful use as a text, the sole prerequisite being the subject matter of L. E. Dicksons First Course in the Theory of Equations. However, I am fully aware of the serious gap in mode of thought between the intuitive treatment of algebraic theory of the First Course and the rigorous abstract treatment of the Modern Higher Algebra, as well as the pedagogical difficulty which is a consequence. The publication recently of more abstract presentations of the theory of equations gives evidence of attempts to diminish this gap. Another such at tempt has resulted in a supposedly less abstract treatise on modern algebra which is about to appear as these pages are being written. However, I have the feeling that neither of these compromises is desirable and that it would be far better to make the transition from the intuitive to the abstract by the addition of a new course in algebra to the undergraduate curriculum in mathematics a curriculum which contains at most two courses in algebra and these only partly algebraic in content. This book is a text for such a course. In fact, its only prerequisite ma terial is a knowledge of that part of the theory of equations given as a chap ter of the ordinary text in college algebra as well as a reasonably complete knowledge of the theory of determinants. Thus, it would actually be pos sible for a student with adequate mathematical maturity, whose only train ing in algebra is a course in college algebra, to grasp the contents. I have used the text in manuscript form in a class composed of third-and fourth-year undergraduate and beginning graduate students, and they all seemed to find the material easy to understand. I trust that it will find such use elsewhere and that it will serve also to satisfy the great interest in the theory of matrices which has been shown me repeatedly by students of the social sciences. I wish to express my deep appreciation of the fine critical assistance of Dr. Sam Perils during the course of publication of this book. UNIVERSITY OF CHICAGO A. A. ALBERT September 9, 1940 v TABLE OF CONTENTS CHAPTER PAQB I. POLYNOMIALS 1 1. Polynomials in x 1 2. The division algorithm 4 3. Polynomial divisibility 5 4. Polynomials in several variables 6 5. Rational functions 8 6. A greatest common divisor process 9 7. Forms 13 8. Linear forms 15 9. Equivalence of forms 17 II. RECTANGULAR MATRICES AND ELEMENTARY TRANSFORMATIONS . . 19 1. The matrix of a system of linear equations 19 2. Submatrices 21 3. Transposition 22 4. Elementary transformations 24 5. Determinants 26 6. Special matrices 29 7. Rational equivalence of rectangular matrices 32 III. EQUIVALENCE OF MATRICES AND OF FORMS 36 1. Multiplication of matrices 36 2. The associative law 38 3. Products by diagonal and scalar matrices - . 39 4. Elementary transformation matrices 42 5. The determinant of a product 44 6. Nonsingular matrices 45 7. Equivalence of rectangular matrices 47 8. Bilinear forms 48 9. Congruence of square matrices 51 10. Skew matrices and skew bilinear forms 52 11. Symmetric matrices and quadratic forms 53 12. Nonmodular fields 56 13. Summary of results 58 14. Addition of matrices 59 15. Real quadratic forms 62 IV...