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Algebra - A Graduate Course (Graduate Studies in Mathematics)
By I. Martin Isaacs
American Mathematical Society | January 2009 | ISBN-10: 0821847996 | DJVU/PDF | 516 pages | 8.3/15.8 mb
http://www.amazon.com/Algebra-Graduate-Studies-Mathematics-Martin/dp/0821847996
PDF conversion is mine.
This book, based on a first-year graduate course the author taught at the University of Wisconsin, contains more than enough material for a two-semester graduate-level abstract algebra course, including groups, rings and modules, fields and Galois theory, an introduction to algebraic number theory, and the rudiments of algebraic geometry. In addition, there are some more specialized topics not usually covered in such a course. These include transfer and character theory of finite groups, modules over artinian rings, modules over Dedekind domains, and transcendental field extensions. This book could be used for self study as well as for a course text, and so full details of almost all proofs are included, with nothing being relegated to the chapter-end problems. There are, however, hundreds of problems, many being far from trivial. The book attempts to capture some of the informality of the classroom, as well as the excitement the author felt when taking the corresponding course as a student.
Contents
PART ONE Noncommutative Algebra 1
CHAPTER 1 Definitions and Examples o[Groups 3
CHAPTER 2 Subgroups and Cosets 14
CHAPTER 3 Homomorphisms 30
CHAPTER 4 Group Actions 42
CHAPTER 5 The Sylow Theorems and p-groups 55
CHAPTER 6 Permutation Groups 70
CHAPTER 7 New Groups from Old 83
CHAPTER 8 Solvable and Nilpotent Groups 99
CHAPTER 9 Transfer lJ 5
CHAPTER 10 Operator Groups and Unique Decompositions 129
CHAPTER I I Module Theory without Rings 142
CHAPTER 12 Rings, Ideals, and Modules 159
CHAPTER 13 Simple Modules and Primitive Rings 177
CHAPTER 14 Artinian Rings and Projective Modules 194
CHAPTER 15 An Introduction to Character Theory 213
PART TWO Commutative Algebra 231
CHAPTER 16 Polynomial Rings, P/Ds, and UFDs 233
CHAPTER 17 Field Extensions 254
CHAPTER 18 Galois Theory 274
CHAPTER 19 Separability and Inseparability 293
CHAPTER 20 Cyclotomy and Geometric Constructions 307
CHAPTER 21 Finite Fields 326
CHAPTER 22 Roots, Radicals, and Real Numbers 342
CHAPTER 23 Norms, Traces, and Discriminants 359
CHAPTER 24 Transcendental Extensions 379
CHAPTER 25 The Artin-Schreier Theorem 401
CHAPTER 26 Ideal Theory 418
CHAPTER 27 Noetherian Rings 433
CHAPTER 28 Integrality 453
CHAPTER 29 Dedekind Domains 474
CHAPTER 30 Algebraic Sets and the Nullstellensatz 493
Index 507
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