Algebra: A Graduate Course available in Hardcover
- Pub. Date:
- American Mathematical Society
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.
Table of ContentsPART 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