Advanced Modern Algebra - math.hcmuns.edu.vn

1 Advanced Modern Algebra by Joseph J. Rotman Hardcover: 1040 pages Publisher: Prentice Hall; 1st edition (2002); 2nd printing (2003). Language: English ISBN: 0130878685. Book Description This book's organizing principle is the interplay between groups and rings, where rings includes the ideas of modules. It contains basic definitions, complete and clear theorems (the first with brief sketches of proofs), and gives attention to the topics of algebraic geometry, computers, homology, and representations. More than merely a succession of definition-theorem-proofs, this text put results and ideas in context so that students can appreciate why a certain topic is being studied, and where definitions originate.

2 Chapter topics include groups; commutative rings; modules; principal ideal domains;. algebras; cohomology and representations; and homological Algebra . For individuals interested in a self-study guide to learning Advanced Algebra and its related topics. Book Info Contains basic definitions, complete and clear theorems, and gives attention to the topics of algebraic geometry, computers, homology, and representations. For individuals interested in a self-study guide to learning Advanced Algebra and its related topics. To my wife Marganit and our two wonderful kids, Danny and Ella, whom I love very much Contents Second Printing.

3 Viii Preface .. ix Etymology .. xii Special Notation .. xiii Chapter 1 Things Past .. 1. Some Number Theory .. 1. Roots of Unity .. 15. Some Set Theory .. 25. Chapter 2 Groups I .. 39. Introduction .. 39. Permutations .. 40. Groups .. 51. Lagrange's Theorem .. 62. Homomorphisms .. 73. Quotient Groups .. 82. Group Actions .. 96. Chapter 3 Commutative Rings I .. 116. Introduction .. 116. First Properties .. 116. Polynomials .. 126. Greatest Common Divisors .. 131. Homomorphisms .. 143. Euclidean Rings .. 151. Linear Algebra .

4 158. Vector Spaces .. 159. Linear Transformations .. 171. Quotient Rings and Finite Fields .. 182. v vi Contents Chapter 4 Fields .. 198. Insolvability of the Quintic .. 198. Formulas and Solvability by Radicals .. 206. Translation into Group Theory .. 210. Fundamental Theorem of Galois Theory .. 218. Chapter 5 Groups II .. 249. Finite Abelian Groups .. 249. Direct Sums .. 249. Basis Theorem .. 255. Fundamental Theorem .. 262. The Sylow Theorems .. 269. The Jordan H older Theorem .. 278. Projective Unimodular Groups .. 289.

5 Presentations .. 297. The Nielsen Schreier Theorem .. 311. Chapter 6 Commutative Rings II .. 319. Prime Ideals and Maximal Ideals .. 319. Unique Factorization Domains .. 326. Noetherian Rings .. 340. Applications of Zorn's Lemma .. 345. Varieties .. 376. Gr obner Bases .. 399. Generalized Division Algorithm .. 400. Buchberger's Algorithm .. 411. Chapter 7 Modules and Categories .. 423. Modules .. 423. Categories .. 442. Functors .. 461. Free Modules, Projectives, and Injectives .. 471. Grothendieck Groups .. 488. Limits .. 498.

6 Chapter 8 Algebras .. 520. Noncommutative Rings .. 520. Chain Conditions .. 533. Semisimple Rings .. 550. Tensor Products .. 574. Characters .. 605. Theorems of Burnside and of Frobenius .. 634. Contents vii Chapter 9 Advanced Linear Algebra .. 646. Modules over PIDs .. 646. Rational Canonical Forms .. 666. Jordan Canonical Forms .. 675. Smith Normal Forms .. 682. Bilinear Forms .. 694. Graded Algebras .. 714. Division Algebras .. 727. Exterior Algebra .. 741. Determinants .. 756. Lie Algebras .. 772. Chapter 10 Homology .. 781.

7 Introduction .. 781. Semidirect Products .. 784. General Extensions and Cohomology .. 794. Homology Functors .. 813. Derived Functors .. 830. Ext and Tor .. 852. Cohomology of Groups .. 870. Crossed Products .. 887. Introduction to Spectral Sequences .. 893. Chapter 11 Commutative Rings III .. 898. Local and Global .. 898. Dedekind Rings .. 922. Integrality .. 923. Nullstellensatz Redux .. 931. Algebraic Integers .. 938. Characterizations of Dedekind Rings .. 948. Finitely Generated Modules over Dedekind Rings .. 959. Global Dimension.

8 969. Regular Local Rings .. 985. Appendix The Axiom of Choice and Zorn's Lemma .. A-1. Bibliography .. B-1. Index .. I-1. Second Printing It is my good fortune that several readers of the first printing this book apprised me of errata I had not noticed, often giving suggestions for improvement. I give special thanks to Nick Loehr, Robin Chapman, and David Leep for their generous such help. Prentice Hall has allowed me to correct every error found; this second printing is surely better than the first one. Joseph Rotman May 2003.

9 Viii Preface Algebra is used by virtually all mathematicians, be they analysts, combinatorists, com- puter scientists, geometers, logicians, number theorists, or topologists. Nowadays, ev- eryone agrees that some knowledge of linear Algebra , groups, and commutative rings is necessary, and these topics are introduced in undergraduate courses. We continue their study. This book can be used as a text for the first year of graduate Algebra , but it is much more than that. It can also serve more Advanced graduate students wishing to learn topics on their own; while not reaching the frontiers, the book does provide a sense of the successes and methods arising in an area.

10 Finally, this is a reference containing many of the standard theorems and definitions that users of Algebra need to know. Thus, the book is not only an appetizer, but a hearty meal as well. When I was a student, Birkhoff and Mac Lane's A Survey of Modern Algebra was the text for my first Algebra course, and van der Waerden's Modern Algebra was the text for my second course. Both are excellent books (I have called this book Advanced Modern Algebra in homage to them), but times have changed since their first appearance: Birkhoff and Mac Lane's book first appeared in 1941, and van der Waerden's book first appeared in 1930.