Advanced Modern Algebra

Advanced Modern Algebra

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For two-semester, beginning graduate-level courses in Algebra.The new "Bibles of Graduate 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 proofs, and gives attention to the topics of algebraic geometry, Grobner bases, 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.
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Product details

  • Hardback | 1040 pages
  • 184 x 234 x 42mm | 1,678.31g
  • Pearson
  • United States
  • English
  • 0130878685
  • 9780130878687

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.
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