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Advanced Modern Algebra
Joseph J. Rotman, University of Illinois, Urbana-Champaign

ISBN-10: 0130878685
ISBN-13: 9780130878687

Publisher: Prentice Hall
Copyright: 2002
Format: Cloth; 1040 pp
Published: 04/25/2002
Status: Instock

Suggested retail price: $134.67
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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, Gröbner 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.

Coverage of topics not usually found in other texts—e.g., inverse and direct limits; Euclidean rings; Gröbner bases; Ext and Tor; Schreier-Neilsen theorem (subgroups of free groups are free); simplicity of PSL (2, q ); etc.
- Gives students a much broader understanding of modern abstract algebra. Ex.___

Many examples and counter-examples and exercises.
- Give students practice in learning to understand concepts. Ex.___

Applications of Zorn's lemma—Include I. S. Cohen's theorem that a commutative ring is noetherian if and only if every prime ideal in it is finitely generated; existence and uniqueness of algebraic closures; transcendence degree; maximal separable extensions.

Serious treatment of set theory—Reminds students what functions really are.
- Enables students to decide when two functions are equal, or describe functions from, or to, the empty set, Equivalence of Zorn's lemma, and the axiom of Choice. Ex.___

Proof of the Fundamental Theorem of Finite Abelian Groups is given in Chapter 5 before the proof of its generalization to finitely generated modules over PIDs in Chapter 9.
- 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. Ex.___

Smooth transition in Chapters 1, 2, and 3.
- Makes 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 readers having different backgrounds will have no difficulty finding an appropriate starting point. Ex.___

Polynomials in several variables.
- Unique factorization, Hilbert basis theorem, Nullstellensatz, irreducible components of affine varieties, primary decomposition. Ex.___

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-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.

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.

Abstract Algebra [GRADUATE] (Mathematics)

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