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Advanced Modern Algebra
Joseph J. Rotman
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-H�lder 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. Gr�bner 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.