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.
I. Martin Isaacs, University of Wisconsin, Madison, WI.
Part One. Noncommutative algebra Chapter 1. Definitions and examples of groups Chapter 2. Subgroups and cosets Chapter 3. Homomorphisms Chapter 4. Group actions Chapter 5. The Sylow theorems and $p$-groups Chapter 6. Permutation groups Chapter 7. New groups from old Chapter 8. Solvable and nilpotent groups Chapter 9. Transfer Chapter 10. Operator groups and unique decompositions Chapter 11. Module theory without rings Chapter 12. Rings, ideals, and modules Chapter 13. Simple modules and primitive rings Chapter 14. Artinian rings and projective modules Chapter 15. An introduction to character theory Part Two. Commutative algebra Chapter 16. Polynomial rings, PIDs, and UFDs Chapter 17. Field extensions Chapter 18. Galois theory Chapter 19. Separability and inseparability Chapter 20. Cyclotomy and geometric constructions Chapter 21. Finite fields Chapter 22. Roots, radicals, and real numbers Chapter 23. Norms, traces, and discriminants Chapter 24. Transcendental extensions Chapter 25. the Artin-Schreier theorem Chapter 26. Ideal theory Chapter 27. Noetherian rings Chapter 28. Integrality Chapter 29. Dedekind domains Chapter 30. Algebraic sets and the Nullstellensatz
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
