ISBN13: 978-0130878687

ISBN10: 0130878685

Cover type:

Edition/Copyright: 02

Publisher: Prentice Hall, Inc.

Published: 2002

International: No

ISBN10: 0130878685

Cover type:

Edition/Copyright: 02

Publisher: Prentice Hall, Inc.

Published: 2002

International: No

Author Bio

**Rotman, Joseph J. : University of Illinois, Urbana-Champaign **

Preface.

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. Epilog.

4. Fields.

Introduction. Insolvability of the Quintic. Fundamental Theorem of Galois Theory.

5. Groups II.

Finite Abelian Groups. The Sylow Theorems. The Jordan-Hölder Theorem. PSL (2,k). 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. The Jacobson Radical. Semisimple Rings. Tensor Products. Representations and Burnside's Theorem.

9. Advanced Linear Algebra.

Modules over PIDs. Rational Canonical Forms. Jordan Canonical Forms. Smith Normal Forms. Bilinear Forms. Graded Algebras. Exterior Algebra. Determinants. Lie Algebras.

10. Homology.

Introduction. Semidirect Products. General Extensions and Cohomology. Homology Functors. Ext and Tor. Cohomology of Groups. Epilog.

11. Commutative Rings III.

Dedekind Rings. Global Dimension. Local Rings. Regular Local Rings.

Appendix A: The Axiom of Choice and Zorn's Lemma.

Bibliography.

Index.

ISBN10: 0130878685

Cover type:

Edition/Copyright: 02

Publisher: Prentice Hall, Inc.

Published: 2002

International: No

Author Bio

**Rotman, Joseph J. : University of Illinois, Urbana-Champaign **

Table of Contents

Preface.

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. Epilog.

4. Fields.

Introduction. Insolvability of the Quintic. Fundamental Theorem of Galois Theory.

5. Groups II.

Finite Abelian Groups. The Sylow Theorems. The Jordan-Hölder Theorem. PSL (2,k). 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. The Jacobson Radical. Semisimple Rings. Tensor Products. Representations and Burnside's Theorem.

9. Advanced Linear Algebra.

Modules over PIDs. Rational Canonical Forms. Jordan Canonical Forms. Smith Normal Forms. Bilinear Forms. Graded Algebras. Exterior Algebra. Determinants. Lie Algebras.

10. Homology.

Introduction. Semidirect Products. General Extensions and Cohomology. Homology Functors. Ext and Tor. Cohomology of Groups. Epilog.

11. Commutative Rings III.

Dedekind Rings. Global Dimension. Local Rings. Regular Local Rings.

Appendix A: The Axiom of Choice and Zorn's Lemma.

Bibliography.

Index.

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