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內(nèi)容概要

　　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.

書籍目錄

Part One The Basic Objects of Algebra Chapter 1 Groups 1. Monoids 2. Groups 3. Normal subgroups 4. Cyclic groups5. 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 Rings1. Rings and homomorphisms 2. Commutative rings 3. Polynomials and group rings 4. Localization 5. Principal and factorial ringsChapter 3 ModulesChapter 4 PolynomlalsPart Two Algebraic EquationsChapter 5 Algebralc ExtensionsChapter 6 Galois TheoryChapter 7 Extensions of RingsChapter 8 Transcendental ExtensionsChapter 9 Algebraic SpacesChapter 10 Noetherial Rings and ModulesChapter 11 Real FieldsChapter 12 Absolute ValuesPart Three Liear Alebar and ReqresentationsChapter 13 Matrices and Linear MapsChapter 14 Representatlon of One EndomorphismChapter 15 Structure of Bilinear FormsChapter 16 The Tensor ProductChapter 17 SmisimplicityChapter 18 Representations of Finite GroupsChapter 19 The Alternating ProductPart Four Homological AlgebraChapter 20 General Homology TheoryChapter 21 Finite Free ResolutionsAppendix 1 The Transcendence of e and Appendix 2 Some Set TheoryBibliographyIndex

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