Description
Book SynopsisWith a focus on one central theme (the Impossibility Theorem) throughout, this highly accessible introduction to Galois theory presents a classical treatment of the topic and poses questions related to the solvability of polynomial equations by radicals. Modern points of view are also discussed in contrast to the historical development and context.
Table of ContentsPreface 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
