伽罗瓦理论

acknowledgments xiii

1. galois 2. influence of lagrange 3. quadratic equations 4.1700 n.c. to a.o. 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. lagrange and vandermonde 16. lagrange resolvents 17. solution of quartic again 18. attempt at quintic ~19.lagrange's rdfiexions

second exercise set

20. cyciotomic 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 gaiois 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 ofindex 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 offactorization 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 galois 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 of24

eighth exercise set

appendix 1. memoir on the conditions for solvability of equations by radicals, by evariste galois

appendix 2. synopsis

appendix 3. groups

This exposition of Galois theory was originally going to be Chapter I of the continuation of my book Ferrnat's Last Theorem, but it soon outgrew any reasonable bounds for an introductory chapter, and I decided to make it a separate book. However, this decision was prompted by more than just the length. Following the precepts of my sermon "Read the Masters!" , Imade the reading of Galois' original memoir a major part of my study of Galois theory, and I saw that the modern treatments of Galois theory lacked much of the simplicity and clarity of the original. Therefore I wanted to write about the theory in a way that would not only explain it, but explain it in terms close enough to Galois' own to make his memoir accessible to the reader, in the same way that I tried to make Riemann's memoir on the zeta function and Kummer's papers on Fermat's Last Theorem accessible in my earlier books, . Clearly I could not do this within the confines of one expository chapter