伽羅瓦理論

內(nèi)容概要

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!" [E2], 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, [Eli and [E3]. Clearly I could not do this within the confines of one expository chapter

書籍目錄

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

章節(jié)摘錄

　　Great mathematicians usually have undramatic lives， or， more pre-cisely， the drama of their lives lies in their mathematics and cannot be appreci-ated by nonmathematicians. The great exception to this rule is Evariste Galois（1811-1832）. Galois life story——what we know of it——is like a romantic novel.Although he was making important mathematical discoveries when he wasstill in secondary school， he was denied admission to the Ecole Polytechnique，which was the premier institution of higher learning in mathematics at thetime， and the mathematical establishment ignored， mislaid， lost， and failedto understand his treatises. Meanwhile， he was persecuted for his politicalactivities and spent many months in jail as a political prisoner. At the age of20 he was killed in a duel involving， in some mysterious way， honor and awoman. On the eve of the fatal duel he wrote a letter to a friend outlining hismathematical accomplishments and asking that the friend try to bring hiswork to the attention of the mathematical world. Against great odds， Galoisfew supporters did finally， 14 years after his death， succeed in finding anaudience for his work， and portions of his writings were published in 1846 byJoseph Liouville in his Journal de Mathematiques. After that， recognition ofthe great importance of his work came very quickly， and Galois began to beregarded， as he is today， as one of the great creative mathematicians of alltime.

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