Classical Galois theory: with examples

PREFACE......Page 5
CHAPTER I PREREQUISITES......Page 9
1.1 Group Theory......Page 9
1.2 Permutations and Permutation Groups......Page 15
1.3 Fields......Page 20
1.4 Rings and Polynomials......Page 22
1.5 Some Elementary Theory of Equations......Page 30
1.6 Vector Spaces......Page 37
CHAPTER II FIELDS......Page 41
2.1 Degree of an Algebraic Extension......Page 41
2.2 Isomorphisms of Fields......Page 47
2.3 Automorphisms of Fields......Page 58
2.4 Fixed Fields......Page 65
CHAPTER III FUNDAMENTAL THEOREM......Page 77
3.1 Splitting Fields......Page 77
3.2 Normal Extensions and Groups of Automorphisms......Page 84
3.3 Conjugate Fields and Elements......Page 96
3.4 Fundamental Theorem......Page 101
CHAPTER IV APPLICATIONS......Page 129
4.1 Solvability of Equations......Page 129
4.2 Solvable Equations Have Solvable Groups......Page 133
4.3 General Equation of Degree n......Page 144
4.4 Roots of Unity and Cyclic Equations......Page 147
4.5 How to Solve a Solvable Equation......Page 165
4.6 Ruler-and-Compass Constructions......Page 188
4.7 Lagranges Theorem......Page 204
4.8 Resolvent of a Polynomial......Page 210
4.9 Calculation of the Galois Group......Page 220
4.10 Matrix Solutions of Equations......Page 227
4.11 Finite Fields......Page 230
4.12 More Applications......Page 241
Bibliography......Page 253
Index......Page 255