A Classical Introduction to Galois Theory

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thumbnail image: A Classical Introduction to Galois Theory

Explore the foundations and modern applications of Galois theory

Galois 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.

PREFACE xi

1 CLASSICAL FORMULAS 1

1.1 Quadratic Polynomials / 3

1.2 Cubic Polynomials / 5

1.3 Quartic Polynomials / 11

2 POLYNOMIALS AND FIELD THEORY 15

2.1 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 41

3.1 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 65

4.1 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 80

5.1 Roots of Unity / 80

5.2 Cyclotomic Polynomials / 82

6 RADICAL EXTENSIONS AND SOLVABILITY BY RADICALS 89

6.1 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 117

7.1 General Polynomials / 117

7.2 The Beginnings of Galois Theory / 124

8 CLASSICAL GALOIS THEORY ACCORDING TO GALOIS 135

9 MODERN GALOIS THEORY 151

9.1 Galois Theory and Finite Extensions / 152

9.2 Galois Theory and Splitting Fields / 156

10 CYCLIC EXTENSIONS AND CYCLOTOMIC FIELDS 171

10.1 Cyclic Extensions / 171

10.2 Cyclotomic Fields / 179

11 GALOIS’S CRITERION FOR SOLVABILITY OF POLYNOMIALS BY RADICALS 185

12 POLYNOMIALS OF PRIME DEGREE 192

13 PERIODS OF ROOTS OF UNITY 200

14 DENESTING RADICALS 225

15 CLASSICAL FORMULAS REVISITED 231

15.1 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

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