Advanced Algebra / Edition 1by Joseph J. Rotman
Pub. Date: 04/30/2002
Publisher: Prentice Hall
This book's organizing principle is the interplay between groups and rings, where “rings” includes the ideas of modules. It contains basic definitions, complete and clear theorems (the first with brief sketches of proofs), and gives attention to the topics of algebraic geometry, computers, homology, and representations. More than merely a succession of
This book's organizing principle is the interplay between groups and rings, where “rings” includes the ideas of modules. It contains basic definitions, complete and clear theorems (the first with brief sketches of proofs), and gives attention to the topics of algebraic geometry, computers, homology, and representations. More than merely a succession of definition-theorem-proofs, this text put results and ideas in context so that students can appreciate why a certain topic is being studied, and where definitions originate.
Chapter topics include groups; commutative rings; modules; principal ideal domains; algebras; cohomology and representations; and homological algebra.
For individuals interested in a self-study guide to learning advanced algebra and its related topics.
- Prentice Hall
- Publication date:
- Edition description:
- Older Edition
- Product dimensions:
- 7.10(w) x 9.50(h) x 1.60(d)
Table of Contents
1. Things Past.
Some Number Theory. Roots of Unity. Some Set Theory.
2. Groups I.
Introduction. Permutations. Groups. Lagrange's Theorem. Homomorphisms. Quotient Groups. Group Actions.
3. Commutative Rings I.
Introduction. First Properties. Polynomials. Greatest Common Divisors. Homomorphisms. Euclidean Rings. Linear Algebra. Quotient Rings and Finite Fields.
Insolvability of the Quintic. Fundamental Theorem of Galois Theory.
5. Groups II.
Finite Abelian Groups. The Sylow Theorems. The Jordan-Hölder Theorem. Projective Unimodular Groups. Presentations. The Neilsen-Schreier Theorem.
6. Commutative Rings II.
Prime Ideals and Maximal Ideals. Unique Factorization Domains. Noetherian Rings. Applications of Zorn's Lemma. Varieties. Gröbner Bases.
7. Modules and Categories.
Modules. Categories. Functors. Free Modules, Projectives, and Injectives. Limits.
Noncommutative Rings. Chain Conditions. Semisimple Rings. Tensor Products. Characters. Theorems of Burnside and Frobenius.
9. Advanced Linear Algebra.
Modules over PIDs. Rational Canonical Forms. Jordan Canonical Forms. Smith Normal Forms. Bilinear Forms. Graded Algebras. Division Algebras. Exterior Algebra. Determinants. Lie Algebras.
Introduction. Semidirect Products. General Extensions and Cohomology. Homology Functors. Derviced Functors. Ext and Tor. Cohomology of Groups. Crossed Products. Introduction to Spectral Sequences.
11. Commutative Rings III.
Local and Global. Dedekind Rings. Global Dimension. Regular Local Rings.
Appendix A: The Axiom of Choice and Zorn's Lemma.
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