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**Algebra** fulfills a definite need to provide a self-contained, one volume, graduate level algebra text that is readable by the average graduate student and flexible enough to accomodate a wide variety of instructors and course contents. The guiding philosophical principle throughout the text is that the material should be presented in the maximum usable generality consistent with good pedagogy. Therefore it is essentially self-contained, stresses clarity rather than brevity and contains an unusually large number of illustrative exercises. The book covers major areas of modern algebra, which is a necessity for most mathematics students in sufficient breadth and depth. Cover 1 Title 3 Copyrigth Page 6 Preface to the Springer Edition 9 Preface 11 Acknowledgments 15 Suggestions on the Use of this Book 17 Table of Contents 23 Introduction: Prerequisites and Preliminaries 27 1. Logic 27 2. Sets and Classes 27 3. Functions 29 4. Relations and Partitions 32 5. Products 33 6. The Integers 35 7. The Axiom of Choice, Order and Zorn's Lemma 38 8. Cardinal Numbers 41 Chapter I: Groups 49 1. Semigroups, Monoids and Groups 50 2. Homomorphisms and Subgroups 56 3. Cyclic Groups 61 4. Cosets and Counting 63 5. Normality, Quotient Groups, and Homomorphisms 67 6. Symmetric, Alternating, and Dihedral Groups 72 7. Categories: Products, Coproducts, and Free Objects 78 8. Direct Products and Direct Sums 85 9. Free Groups, Free Products, and Generators and Relations 90 Chapter II: The Structure of Groups 96 1. Free Abelian Groups 96 2. Finitely Generated Abelian Groups 102 3. The Krull-Schmidt Theorem 109 4. The Action of a Group on a Set 114 5. The Sylow Theorems 118 6. Classification of Finite Groups 122 7. Nilpotent and Solvable Groups 126 8. Normal and Subnormal Series 133 Chapter III: Rings 140 1. Rings and Homomorphisms 141 2. Ideals 148 3. Factorization in Commutative Rings 161 4. Rings of Quotients and Localization 168 5. Rings of Polynomials and Formal Power Series 175 6. Factorization in Polynomial Rings 183 Chapter IV: Modules 194 1. Modules, Homomorphisms and Exact Sequences 195 2. Free Modules and Vector Spaces 206 3. Projective and lnjective Modules 216 4. Hom and Duality 225 5. Tensor Products 233 6. Modules over a Principal Ideal Domain 244 7. Algebras 252 Chapter V: Fields and Galois Theory 256 1. Field Extensions 257 Appendix: Ruler and Compass Constructions 264 2. The Fundamental Theorem 269 Appendix: Symmetric Rational Functions 278 3. Splitting Fields, Algebraic Closure and Normality 283 Appendix: The Fundamental Theorem of Algebra 291 4. The Galois Group of a Polynomial 295 5. Finite Fields 304 6. Separability 308 7. Cyclic Extensions 315 8. Cyclotomic Extensions 323 9. Radical Extensions 328 Appendix: The General Equation of Degree n 333 Chapter VI: The Structure of Fields 337 1. Transcendence Bases 337 2. Linear Disjointness and Separability 344 Chapter VII: Linear Algebra 353 1.Matrices and Maps 354 2. Rank and Equivalence 361 Appendix: Abelian Groups Defined by Generators and Relations 369 3. Determinants 374 4. Decomposition of a Single Linear Transformation and Similarity 381 5. The Characteristic Polynomial, Eigenvectors and Eigenvalues 392 Chapter VIII: Commutative Rings and Modules 397 1. Chain Conditions 398 2. Prime and Primary Ideals 403 3. Primary Decomposition 409 4. Noetherian Rings and Modules 413 5. Ring Extensions 420 6. Dedekind Domains 426 7. The Hilbert Nullstellensatz 435 Chapter IX: The Structure of Rings 440 1. Simple and Primitive Rings 441 2. The Jacobson Radical 450 3. Semisimple Rings 460 4. The Prime Radical; Prime and Semiprime Rings 470 5. Algebras 476 6. Division Algebras 482 Chapter X: Categories 490 1. Functors and Natural Transformations 491 2. Adjoint Functors 502 3. Morphisms 506 List of Symbols 511 Bibliography 515 Index 519 Finally a self-contained, one volume, graduate-level algebra text that is readable by the average graduate student and flexible enough to accommodate a wide variety of instructors and course contents. The guiding principle throughout is that the material should be presented as general as possible, consistent with good pedagogy. Therefore it stresses clarity rather than brevity and contains an extraordinarily large number of illustrative exercises. In Sections 1-6 we summarize for the reader's convenience some basic material with which he is assumed to be thoroughly familiar (with the possible exception of the distinction between sets and proper classes (Section 2), the characterization of the Cartesian product by a universal mapping property (Theorem 5.2) and the Recursion Theorem 6.2).