**Graduate Studies in Mathematics**

Volume: 100;
1994;
516 pp;
Hardcover

MSC: Primary 00;
Secondary 12; 13; 16; 20

**Print ISBN: 978-0-8218-4799-2
Product Code: GSM/100**

List Price: $88.00

AMS Member Price: $70.40

MAA Member Price: $79.20

**Electronic ISBN: 978-1-4704-1164-0
Product Code: GSM/100.E**

List Price: $83.00

AMS Member Price: $66.40

MAA Member Price: $74.70

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#### Supplemental Materials

# Algebra: A Graduate Course

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*I. Martin Isaacs*

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.

Originally published by Brooks Cole/Cengage Learning as ISBN:
978-0-534-19002-6

#### Readership

Graduate students and research mathematicians interested in algebra.

#### Reviews & Endorsements

Unlike similar textbooks, this volume steers away from chapter-end problems by including full details of all proofs as problems are presented.

-- SciTech Book News

This is a book to be warmly welcomed. The presentation throughout is a model of clarity, and the proofs are precise and complete. The careful reader will learn (much) from it, not only mathematics, but also (and more importantly) how to think mathematically.

-- Mathematical Reviews

Isaacs' *Algebra,
A Graduate Course* is a pedagogically important book, to be highly
recommended to fledgling algebraists—and every one else, for
that matter.

-- MAA Reviews

Most of these extra topics are not usually covered in first-year graduate algebra courses, or in introductory textbooks on modern algebra, but here they are woven into the main text in very natural, effective and instructive a manner, thereby offering a wider panorama of abstract algebra to the interested reader. This profound algebra text will prepare any zealous reader for further steps into one or more of the many branches of algebra, algebraic number theory, or algebraic geometry. Also, it will maintain its well-established role as one of the excellent standard texts on the subject, as a highly recommendable source for instructors, and as an utmost valuable companion to the various other great textbooks in the field.

-- Zentralblatt MATH

#### Table of Contents

# Table of Contents

## Algebra: A Graduate Course

- Cover Cover11 free
- Title i2 free
- Copyright ii3 free
- Contents ix10 free
- PART ONE: Noncommutative Algebra 114 free
- CHAPTER 1. Definitions and Examples of Groups 316
- CHAPTER 2. Subgroups and Cosets 1427
- CHAPTER 3. Homomorphisms 3043
- CHAPTER 4. Group Actions 4255
- CHAPTER 5. The Sylow Theorems and p-groups 5568
- CHAPTER 6. Permutation Groups 7083
- CHAPTER 7. New Groups from Old 8396
- CHAPTER 8. Solvable and Nilpotent Groups 99112
- CHAPTER 9. Transfer 115128
- CHAPTER 10. Operator Groups and Unique Decompositions 129142
- CHAPTER 11. Module Theory without Rings 142155
- CHAPTER 12. Rings, Ideals, and Modules 159172
- CHAPTER 13. Simple Modules and Primitive Rings 177190
- CHAPTER 14. Artinian Rings and Projective Modules 194207
- CHAPTER 15. An Introduction to Character Theory 213226

- PART TWO: Commutative Algebra 231244
- CHAPTER 16. Polynomial Rings, PIDs, and UFDs 233246
- CHAPTER 17. Field Extensions 254267
- CHAPTER 18. Galois Theory 274287
- CHAPTER 19. Separability and Inseparability 293306
- CHAPTER 20. Cyclotomy and Geometric Constructions 307320
- CHAPTER 21. Finite Fields 326339
- CHAPTER 22. Roots, Radicals, and Real Numbers 342355
- CHAPTER 23. Norms, Traces, and Discriminants 359372
- CHAPTER 24. Transcendental Extensions 379392
- CHAPTER 25. The Artin-Schreier Theorem 401414
- CHAPTER 26. Ideal Theory 418431
- CHAPTER 27. Noetherian Rings 433446
- CHAPTER 28. Integrality 453466
- CHAPTER 29. Dedekind Domains 474487
- CHAPTER 30. Algebraic Sets and the Nullstellensatz 493506

- Index 507520
- Back Cover Back Cover1530