Synopses & Reviews
This book provides an introduction to Galois theory and focuses on one central theme - the solvability of polynomials by radicals. Both classical and modern approaches to the subject are described in turn in order to have the former (which is relatively concrete and computational) provide motivation for the latter (which can be quite abstract). The theme of the book is historically the reason that Galois theory was created, and it continues to provide a platform for exploring both classical and modern concepts. This book examines a number of problems arising in the area of classical mathematics, and a fundamental question to be considered is: For a given polynomial equation (over a given field), does a solution in terms of radicals exist? That the need to investigate the very existence of a solution is perhaps surprising and invites an overview of the history of mathematics. The classical material within the book includes theorems on polynomials, fields, and groups due to such luminaries as Gauss, Kronecker, Lagrange, Ruffini and, of course, Galois. These results figured prominently in earlier expositions of Galois theory, but seem to have gone out of fashion. This is unfortunate since, aside from being of intrinsic mathematical interest, such material provides powerful motivation for the more modern treatment of Galois theory presented later in the book. Over the course of the book, three versions of the Impossibility Theorem are presented: the first relies entirely on polynomials and fields, the second incorporates a limited amount of group theory, and the third takes full advantage of modern Galois theory. This progression through methods that involve more and more group theory characterizes the first part of the book. The latter part of the book is devoted to topics that illustrate the power of Galois theory as a computational tool, but once again in the context of solvability of polynomial equations by radicals.
Synopsis
Explore the foundations and modern applications of Galois theoryGalois theory is widely regarded as one of the most elegant areas of mathematics. A Classical Introduction to Galois Theory develops the topic from a historical perspective, with an emphasis on the solvability of polynomials by radicals. The book provides a gradual transition from the computational methods typical of early literature on the subject to the more abstract approach that characterizes most contemporary expositions.
The author provides an easily-accessible presentation of fundamental notions such as roots of unity, minimal polynomials, primitive elements, radical extensions, fixed fields, groups of automorphisms, and solvable series. As a result, their role in modern treatments of Galois theory is clearly illuminated for readers. Classical theorems by Abel, Galois, Gauss, Kronecker, Lagrange, and Ruffini are presented, and the power of Galois theory as both a theoretical and computational tool is illustrated through:
- A study of the solvability of polynomials of prime degree
- Development of the theory of periods of roots of unity
- Derivation of the classical formulas for solving general quadratic, cubic, and quartic polynomials by radicals
Throughout the book, key theorems are proved in two ways, once using a classical approach and then again utilizing modern methods. Numerous worked examples showcase the discussed techniques, and background material on groups and fields is provided, supplying readers with a self-contained discussion of the topic.
A Classical Introduction to Galois Theory is an excellent resource for courses on abstract algebra at the upper-undergraduate level. The book is also appealing to anyone interested in understanding the origins of Galois theory, why it was created, and how it has evolved into the discipline it is today.
About the Author
STEPHEN C. NEWMAN, MD, MSc, is Professor Emeritus of Psychiatry at the University of Alberta, Canada. He has published widely in psychiatric epidemiology and epidemiologic methods. Dr. Newman is the author of Biostatistical Methods in Epidemiology (Wiley).
Table of Contents
PREFACE xi
1 CLASSICAL FORMULAS 11.1 Quadratic Polynomials 3
1.2 Cubic Polynomials 5
1.3 Quartic Polynomials 11
2 POLYNOMIALS AND FIELD THEORY 152.1 Divisibility 16
2.2 Algebraic Extensions 24
2.3 Degree of Extensions 25
2.4 Derivatives 29
2.5 Primitive Element Theorem 30
2.6 Isomorphism Extension Theorem and Splitting Fields 35
3 FUNDAMENTAL THEOREM ON SYMMETRIC POLYNOMIALS AND DISCRIMINANTS 413.1 Fundamental Theorem on Symmetric Polynomials 41
3.2 Fundamental Theorem on Symmetric Rational Functions 48
3.3 Some Identities Based on Elementary Symmetric Polynomials 50
3.4 Discriminants 53
3.5 Discriminants and Subfields of the Real Numbers 60
4 IRREDUCIBILITY AND FACTORIZATION 654.1 Irreducibility Over the Rational Numbers 65
4.2 Irreducibility and Splitting Fields 69
4.3 Factorization and Adjunction 72
5 ROOTS OF UNITY AND CYCLOTOMIC POLYNOMIALS 805.1 Roots of Unity 80
5.2 Cyclotomic Polynomials 82
6 RADICAL EXTENSIONS AND SOLVABILITY BY RADICALS 896.1 Basic Results on Radical Extensions 89
6.2 Gauss’s Theorem on Cyclotomic Polynomials 93
6.3 Abel’s Theorem on Radical Extensions 104
6.4 Polynomials of Prime Degree 109
7 GENERAL POLYNOMIALS AND THE BEGINNINGS OF GALOIS THEORY 1177.1 General Polynomials 117
7.2 The Beginnings of Galois Theory 124
8 CLASSICAL GALOIS THEORY ACCORDING TO GALOIS 1359 MODERN GALOIS THEORY 1519.1 Galois Theory and Finite Extensions 152
9.2 Galois Theory and Splitting Fields 156
10 CYCLIC EXTENSIONS AND CYCLOTOMIC FIELDS 17110.1 Cyclic Extensions 171
10.2 Cyclotomic Fields 179
11 GALOIS’S CRITERION FOR SOLVABILITY OF POLYNOMIALS BY RADICALS 18512 POLYNOMIALS OF PRIME DEGREE 19213 PERIODS OF ROOTS OF UNITY 20014 DENESTING RADICALS 22515 CLASSICAL FORMULAS REVISITED 23115.1 General Quadratic Polynomial 231
15.2 General Cubic Polynomial 233
15.3 General Quartic Polynomial 236
APPENDIX A COSETS AND GROUP ACTIONS 245
APPENDIX B CYCLIC GROUPS 249
APPENDIX C SOLVABLE GROUPS 254
APPENDIX D PERMUTATION GROUPS 261
APPENDIX E FINITE FIELDS AND NUMBER THEORY 270
APPENDIX F FURTHER READING 274
REFERENCES 277
INDEX 281