Algebra

Preface Ackno w/edgmen ts Suggestions on the Use of This Book Introduction: Prerequisites and Preliminaries 1. Logic 2. Sets and Classes 3. Functions 4. Relations and Partitions 5. Products 6. The Integers 7. The Axiom of Choice, Order and Zorn's Lemma 8. Cardinal Numbers Chapter I: Groups 1. Semigroups, Monoids and Groups 2. Homomorphisms and Subgroups 3. Cyclic Groups 4. Cosets and Counting 5. Normality, Quotient Groups, and Homomorphisms 6. Symmetric, Alternating, and Dihedral Groups 7. Categories: Products, Coproducts, and Free Objects 8. Direct Products and Direct Sums 9. Free Groups, Free Products, Generators & Relations Chapter I1: The Structure of Groups l. Free Abelian Groups 2. Finitely Generated Abelian Groups 3. The KruU-Schmidt Theorem 4. The Action of a Group on a Set 5. The Sylow Theorems 6. Classification of Finite Groups 7. Nilpotent and Solvable Groups 8. Normal and Subnormal Series Chapter ¢ó: Rings 1. Rings and Homomorphisms 2. Ideals 3. Factorization in Commutative Rings 4. Rings of Quotients and Localization 5. Rings of Polynomials and Formal Power Series 6. Factorization in Polynomial Rings Chapter IV: Modules 1. Modules, Homomorphisms and Exact Sequences 2. Free Modules and Vector Spaces 3. Projective and lnjective Modules 4. Horn and Duality 5. Tensor Products 6. Modules over a Principal Ideal Domain 7. Algebras Chapter V: Fields and Galois Theory I. Field Extensions Appendix: Ruler and Compass Constructions 2. The Fundamental Theorem Appendix: Symmetric Rational Functions 3. Splitting Fields, Algebraic Closure and Normality Appendix: The Fundamental Theorem of Algebra 4. The Galois Group of a Polynomial 5. Finite Fields 6. Separability: 7. Cyclic Extensions 8. Cyclotomic Extensions 9. Radical Extensions Appendix: The General Equation of Degree n Chapter ¢ö: The Structure of Fields 1. Transcendence Bases 2. Linear Disjointness and Separability Chapter ¢÷: Linear Algebra 1.Matrices and Maps 2. Rank and Equivalence Appendix: Abelian Groups Defined by Generators and Relations 3. Determinants 4. Decomposition of a Single Linear Transformation and Similarity. 5. The Characteristic Polynomial, Eigenvectors and "Eigenvalues Chapter ¢ø: Commutative Rings and Modules 1. Chain Conditions 2. Prime and Primary Ideals 3. Primary Decomposition 4. Noetherian Rings and Modules 5. Ring Extensions 6. Dedekind Domains 7. The Hilbert NullsteUensatz Chapter IX: The Structure of Rings 1. Simple and Primitive Rings 2. The Jacobson Radical 3. Semisimple Rings 4. The Prime Radical; Prime and Semiprime Rings 5. Algebras 6. Division Algebras Chapter X: Categories 1. Functors and Natural Transformations 2. Adjoint Functors 3. Morphisms List of Symbols Bibliography index