这本《伽罗瓦理论》由美国的Harold M. Edwards所著，内容是：This exposition of Galois theory was originally going to be Chapter 1 of thecontinuation of my book Fermat's Last Theorem, but it soon outgrew anyreasonable bounds for an introductory chapter, and I decided to make it aseparate book. However, this decision was prompted by more than just thelength. Following the precepts of my sermon "Read the Masters!" [E2], I made the reading of Galois' original memoir a major part of my study ofGalois theory, and I saw that the modern treatments of Galois theory lackedmuch of the simplicity and clarity of the original. Therefore I wanted towrite 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 thereader, in the same way that I tried to make Riemann's memoir on the zetafunction and Kummer's papers on Fermat's Last Theorem accessible in myearlier books, [Eli and I-E3].

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