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Advanced Modern Algebra
by Joseph J. Rotman

Hardcover: 1040 pages Publisher: Prentice Hall; 1st edition (2002); 2nd printing (2003) Language: English ISBN: 0130878685

Book Description This book'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 (the first with briefsketches of proofs), and gives attention to the topics of algebraic geometry, computers, homology, and representations. More than merely a succession of definition-theorem-proofs, this text put results and ideas in context so that students can appreciate why a certain topic is being studied, and where definitions originate. Chapter topics include groups; commutative rings; modules; principal idealdomains; algebras; cohomology and representations; and homological algebra. For individuals interested in a self-study guide to learning advanced algebra and its related topics. Book Info Contains basic definitions, complete and clear theorems, and gives attention to the topics of algebraic geometry, computers, homology, and representations. For individuals interested in a self-study guide to learningadvanced algebra and its related topics.

To my wife Marganit and our two wonderful kids, Danny and Ella, whom I love very much

Contents
Second Printing . Preface . . . . . Etymology . . . Special Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii ix xii xiii 1 1 15 25 39 39 40 51 62 73 82 96

Chapter 1 Things Past . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1. Some Number Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2. Roots of Unity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3. SomeSet Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Chapter 2
2.1. 2.2. 2.3. 2.4. 2.5. 2.6. 2.7.

Groups I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Introduction . . . . . Permutations . . . . . Groups . . . . . . . . Lagrange’s Theorem Homomorphisms . . Quotient Groups . . . Group Actions . . . .

Chapter 3
3.1. 3.2. 3.3. 3.4. 3.5. 3.6.3.7.

Commutative Rings I . . . . . . . . . . . . . . . . . . . . . 116
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 116 126 131 143 151 158 159 171 182

Introduction . . . . . . . . . . . First Properties . . . . . . . . . Polynomials . . . . . . . . . . . Greatest Common Divisors . . . Homomorphisms . . . . . . . . Euclidean Rings . . . . . . . . . Linear Algebra . . . . . .. . . Vector Spaces . . . . . . . . . Linear Transformations . . . . 3.8. Quotient Rings and Finite Fields

v

vi

Contents

Chapter 4 Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
4.1. Insolvability of the Quintic . . . . . . . Formulas and Solvability by Radicals Translation into Group Theory . . . . 4.2. Fundamental Theorem of Galois Theory . . . . . ....
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