Algebra: A Graduate Course / Edition 1

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"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.
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Editorial Reviews

The author encourages students to develop an appreciation of how basic algebra is put together. The text is in two sections: noncommutative algebra, including homomorphisms, Sylow theorems, and rings; and commutative algebra, with polynomial rings, Galois theory, finite fields, Noetherian rings, and Dedekind domains. Discussion of such topic as the factorization algorithm of polynomials over finite fields gives students insight into the types of algorithms that underlie computer algebra software. Problems follow each chapter. Annotation c. Book News, Inc., Portland, OR (
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Product Details

  • ISBN-13: 9780534190026
  • Publisher: Cengage Learning
  • Publication date: 11/28/1993
  • Series: Mathematics Ser.
  • Edition description: New Edition
  • Edition number: 1
  • Pages: 528
  • Product dimensions: 6.53 (w) x 9.48 (h) x 0.92 (d)

Table of Contents

Pt. 1 Noncommutative Algebra 1

Ch. 1 Definitions and Examples of Groups 3

Ch. 2 Subgroups and Cosets 14

Ch. 3 Homomorphisms 30

Ch. 4 Group Actions 42

Ch. 5 The Sylow Theorems and p-groups 55

Ch. 6 Permutation Groups 70

Ch. 7 New Groups from Old 83

Ch. 8 Solvable and Nilpotent Groups 99

Ch. 9 Transfer 115

Ch. 10 Operator Groups and Unique Decompositions 129

Ch. 11 Module Theory without Rings 142

Ch. 12 Rings, Ideals, and Modules 159

Ch. 13 Simple Modules and Primitive Rings 177

Ch. 14 Artinian Rings and Projective Modules 194

Ch. 15 An Introduction to Character Theory 213

Pt. 2 Commutative Algebra 231

Ch. 16 Polynomial Rings, PIDs, and UFDs 233

Ch. 17 Field Extensions 254

Ch. 18 Galois Theory 274

Ch. 19 Separability and Inseparability 293

Ch. 20 Cyclotomy and Geometric Constructions 307

Ch. 21 Finite Fields 326

Ch. 22 Roots, Radicals, and Real Numbers 342

Ch. 23 Norms, Traces, and Discriminants 359

Ch. 24 Transcendental Extensions 379

Ch. 25 The Artin-Schreier Theorem 401

Ch. 26 Ideal Theory 418

Ch. 27 Noetherian Rings 433

Ch. 28 Integrality 453

Ch. 29 Dedekind Domains 474

Ch. 30 Algebraic Sets and the Nullstellensatz 493

Index 507

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