Classical Introduction to Galois Theory (eBook)

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).

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