目錄
Acknowledgments
§1. Galois
§2. Influence of Lagrange
§3. Quadratic equations
§4. 1700 B.c. to A.D. 1500
§5. Solution of cubic
§6. Solution of quartic
§7. Impossibility of quintic
§8. Newton
§9. Symmetric polynomials in roots
§10. Fundamental theorem on symmetric polynomials
§11. Proof
§12. Newton's theorem
§13. Discriminants
First Exercise Set
§14. Solution of cubic
§15. Lagrangc and Vandermondc
§16. Lagrange resolvcnts
§17. Solution of quartic again
§18. Attempt at quintic
§19.Lagrange's Reflexions
Second Exercise Set
§20. Cyclotomic equations
§21. The cases n = 3,5
§22. n = 7,11
§23. General case
§24. Two lemmas
§25. Gauss's method
§26. p-gons by ruler and compass
§27. Summary
Third Exercise Set
§28. Resolvents
§29. Lagrange's theorem
§30. Proof
§31. Galois resolvents
§32. Existence of Galois resolvents
§33. Representation of the splitting field as K(t)
§34. Simple algebraic extensions
§35. Euclidean algorithm
§36. Construction of simple algebraic extensions
§37. Galois' method
Fourth Exercise Set
§38. Review
§39. Finite permutation groups
§40. Subgroups, normal subgroups
§41. The Galois group of an equation
§42. Examples
Fifth Exercise Set
§43. Solvability by radicals
§44. Reduction of the Galois group by a cyclic extension
§45. Solvable groups
§46. Reduction to a normal subgroup of index p
§47. Theorem on solution by radicals (assuming roots of unity)
§48. Summary
Sixth Exercise Set
§49. Splitting fields
§50. Fundamental theorem of algebra (so-called)
§51. Construction of a splitting field
§52. Need for a factorization method
§53. Three theorems on factorization methods
§54. Uniqueness of factorization of polynomials
§55. Factorization over Z
§56. Over Q
§57. Gauss's lemma, factorization over Q
§58. Over transcendental extensions
§59. Of polynomials in two variables
§60. Over algebraic extensions
§61. Final remarks
Seventh Exercise Set
§62. Review of Galois theory
§63. Fundamental theorem of Gaiois theory (so-called)
§64. Galois group of xp - 1 = 0 over Q
§65. Solvability of the cyclotomic equation
§66. Theorem on solution by radicals
§67. Equations with literal coefficients
§68. Equations of prime degree
§69. Galois group of xn - 1 = 0 over Q
§70. Proof of the main proposition
§71. Deduction of Lemma 2 of §24
Eighth Exercise Set
Appendix 1. Memoir on the Conditions for Solvability of Equations by Radicals, by Evariste Galois
Appendix 2. Synopsis
Appendix 3. Groups
Answers to Exercises
List of Exercises
References
Index
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