As I see it, the graduate course in algebra must primarily prepare studentsto handle the algebra which they will meet in all of mathematics: topology,partial differential equations, differential geometry, algebraic geometry, analysis,and representation theory, not to speak of algebra itself and algebraic numbertheory with all its ramifications. Hence I have inserted throughout references topapers and books which have appeared during the last decades, to indicate someof the directions in which the algebraic foundations provided by this book areused; I have accompanied these references with some motivating comments, toexplain how the topics of the present book fit into the mathematics that is tocome subsequently in various fields; and I have also mentioned some unsolvedproblems of mathematics in algebra and number theory. The abc conjecture isperhaps the most spectacular of these.
目录
Part One The Basic Objects of Algebra Chapter 1 Groups 1. Monoids 2. Groups 3. Normal subgroups 4. Cyclic groups 5. Operations of a group on a set 6. Sylow subgroups 7. Direct sums and free abelian groups 8. Finitely generated abelian groups 9. The dual group 10. Inverse limit and completion 11. Categories and functors 12. Free groups Chapter 2 Rings 1. Rings and homomorphisms 2. Commutative rings 3. Polynomials and group rings 4. Localization 5. Principal and factorial rings Chapter 3 Modules Chapter 4 Polynomlals Part Two Algebraic Equations Chapter 5 Algebralc Extensions Chapter 6 Galois Theory Chapter 7 Extensions of Rings Chapter 8 Transcendental Extensions Chapter 9 Algebraic Spaces Chapter 10 Noetherial Rings and Modules Chapter 11 Real Fields Chapter 12 Absolute Values Part Three Liear Alebar and Reqresentations Chapter 13 Matrices and Linear Maps Chapter 14 Representatlon of One Endomorphism Chapter 15 Structure of Bilinear Forms Chapter 16 The Tensor Product Chapter 17 Smisimplicity Chapter 18 Representations of Finite Groups Chapter 19 The Alternating Product Part Four Homological Algebra Chapter 20 General Homology Theory Chapter 21 Finite Free Resolutions Appendix 1 The Transcendence of e and Appendix 2 Some Set Theory Bibliography Index
