Preface Acknowledgments 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 II: The Structure of Groups 1. Free Abelian Groups 2. Finitely Generated Abelian Groups 3. The Krull-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 Ill: 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 Injective Modules 4. Hom and Duality 5. Tensor Products 6. Modules over a Principal Ideal Domain 7. Algebras. Chapter V: Fields and Galois Theory 1. 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 VI: The Structure of Fields Chapter VII: Linear Algebra Chapter VIII: Commutative Rings and Modules Chapter IX: The Structure of Rings Chapter X: Categories List of Symbols Bibliography Index
