Inhaltsverzeichnis
Preface vii
Contents xi
Acknowledgements xiii
Galois
§1. Galois 1
§2. Influence of Lagrange 1
Quadratic Equations 1700 B. C.
§3. Quadratic equations 3
Cubic and Quartic Equations A. D. 1500
§4. 1700 B. C. to A. D. 1500 4
§5. Solution of cubic 4
§6. Solution of quartic 5
§7. Impossibility of quintic 5
Newton and Symmetric Functions
§8. Newton 6
§9. Symmetric polynomials in roots 6
The Fundamental Theorem on Symmetric Polynomials
§10. Fundamental theorem on symmetric polynomials 8
§11. Proof 9
Particular Symmetric Polynomials
§12. Newton's theorem 12
§13. Discriminants 13
First Exercise Set 13
A Method for Solving the Cubic
§14. Solution of cubic 17
Lagrange (Vandermonde) Resolvents
§15. Lagrange and Vandermonde 18
§16. Lagrange resolvents 19
§17. Solution of quartic again 20
§18. Attempt at quintic 21
§19. Lagrange's Réflexions 22
Second Exercise Set 22
Cyclotomic Equations
§20. Cyclotomic equations 23
§21. The cases n = 3, 5 24
Eleventh Roots of Unity
§22. n = 7, 11 24
§23. General case 25
The Cases p > 11
§24. Two lemmas 26
§25. Gauss's method 28
§26. p-gons by ruler and compass 30
Summary
§27. Summary 30
Third Exercise Set 31
Galois Resolvents
§28. Resolvents 32
§29. Lagrange's theorem 33
§30. Proof 34
§31. Galois resolvents 35
§32. Existence of Galois resolvents 36
§33. Representation of the splitting field as K(t) 36
Construction of the Field K(t)
§34. Simple algebraic extensions 37
§35. Euclidean algorithm 39
§36. Construction of simple algebraic extensions 41
Galois' Proof of the Basic Lemma
§37. Galois' method 43
Fourth Exercise Set 45
Basic Galois Theory: The Galois Group
§38. Review 47
§39. Finite permutation groups 48
§40. Subgroups, normal subgroups 50
§41. The Galois group of an equation 51
§42. Examples 54
Fifth Exercise Set 56
Basic Galois Theory: The Groups of Solvable Equations
§43. Solvability by radicals 57
§44. Reduction of the Galois group by a cyclic extension 59
§45. Solvable groups 61
§46. Reduction to a normal subgroup of index p 61
§47. Theorem on solution by radicals (assuming roots of unity) 64
§48. Summary 65
Sixth Exercise Set 65
Roots and Splitting Fields
§49. Splitting fields 67
§50. Fundamental theorem of algebra (so-called) 68
Construction of a Splitting Field
§51. Construction of a splitting field 68
The Need for a Factorization Method
§52. Need for a factorization method 69
§53. Three theorems on factorization methods 70
Unique Factorization into Irreducibles
§54. Uniqueness of factorization of polynomials 71
Factorization Over Q
§55. Factorization over Z 72
§56. Over Q 73
§57. Gauss's lemma, factorization over Q 73
Factorisation Over Transcendental Extensions
§58. Over transcendental extensions 75
§59. Of polynomials in two variables 76
Factorization Over Algebraic Extensions
§60. Over algebraic extensions 76
§61. Final remarks 80
Seventh Exercise Set 81
Review
§62. Review of Galois theory 83
The Fundamental Theorem of Galois Theory
§63. Fundamental theorem of Galois theory (so-called) 84
Construction of pth Roots of Unity
§64. Galois group of xP - 1 = 0 over Q 86
§65. Solvability of the cyclotomic equation 87
Solution by Radicals
§66. Theorem on solution by radicals 88
§67. Equations with literal coefficients 89
Solvable Equations of Prime Degree
§68. Equations of prime degree 91
The Galois Group of xn - 1 = 0
§69. Galois group of xn - 1 = 0 over Q 93
§70. Proof of the main proposition 94
§71. Deduction of Lemma 2 of §24 97
Eighth Exercise Set 97
Appendix 1. Memoir on the Conditions for Solvability of Equations by Radicals, by Evariste Galois 101
Appendix 2. Synopsis 114
Appendix 3. Groups 118
Answers to Exercises 123
List of Exercises 145
References 149
Index 151 - 152