Algebra: A Graduate Course

by I. Martin Isaacs

Overview

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.

Product Details

ISBN-13:	9780821847992
Publisher:	American Mathematical Society
Publication date:	01/13/2009
Series:	Graduate Studies in Mathematics Series , #100
Edition description:	New Edition
Pages:	516
Product dimensions:	6.50(w) x 9.30(h) x 1.20(d)

Table of Contents

PART I: NONCOMMUTATIVE ALGEBRA 1. Definitions and Examples of Groups 2. Subgroups and Cosets 3. Homomorphisms 4. Group Actions 5. The Sylow Theorems and p-groups 6. Permutation Groups 7. New Groups from Old 8. Solvable and Nilpotent Groups 9. Transfer 10. Operator Groups and Unique Decompositions 11. Module Theory without Rings 12. Rings, Ideals, and Modules 13. Simple Modules and Primitive Rings 14. Artinian Rings and Projective Models 15. An Introduction to Character Theory PART II: COMMUTATIVE ALGEBRA 16. Polynomial Rings, PIDs, and UFDs 17. Field Extensions 18. Galois Theory 19. Separability and Inseparability 20. Cyclotomy and Geometric Constructions 21. Finite Fields 22. Roots, Radicals, and Real Numbers 23. Norms, Traces, and Discriminants 24. Transcendental Extensions 25. The Artin-Schreier Theorem 26. Ideal Theory 27. Noetherian Rings 28. Integrality 29. Dedekind Domains 30. Algebraic Sets and the Nullstellensatz Index

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 (booknews.com)

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