For two-term undergraduate level courses in Algebra. This text'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 and gives attention to the topics of algebraic geometry, computers, homology and representations. More than merely a succession of definition theorem proofs, this text puts results and ideas in context so that students can appreciate why a certain topic is being studied and where definitions originate. *Coverage of topics not usually found in other texts - e.g. inverse and direct limits: Euclidean rings; Grobner bases; Ext and tor; Schreier-Neilsen theorem (subgroups of free groups are free); simplicity of PSL (2,q). *Numerous exercises. *Many examples and counter-examples. *Serious treatment of set theory - Reminds students what functions really are. *Early presentation of the basis theorem for finite abelian groups - Makes the proof of the basis theorem for finitely generated modules over PID's more digestible, allowing students to then see how that proof is translated into the language of modules.
*Transition - To make the step from an undergraduate course to this one as smooth as possible, definitions, theorems (with only sketches of proofs) and exercises are presented in Chapter 1 as well as in parts of Chapters 2 and 3, so that the readers having different backgrounds will have no difficulty finding an appropriate starting point. *Polynomials in several variables - Unique factorization, Hilbert basis theorem, Nullstellensatz, irreducible components of affine varieties.
Table of Contents
Preface. Etymology. Special Notation. 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. 4. Fields. Insolvability of the Quintic. Fundamental Theorem of Galois Theory. 5. Groups II. Finite Abelian Groups. The Sylow Theorems. The Jordan-Holder 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. Grobner Bases. 7. Modules and Categories. Modules. Categories. Functors. Free Modules, Projectives, and Injectives. Limits. 8. Algebras. 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. 10. Homology. 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. Bibliography. Index.