Book:Thomas W. Hungerford/Algebra

Subject Matter

 * Set Theory
 * Abstract Algebra

Contents

 * Preface
 * Acknowledgements
 * Suggestions on the Use of This Book


 * Introduction: Prerequisites and Preliminaries


 * Chapter $\text {I}$: 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 $\text {II}$: 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. Classifications of Finite Groups
 * 7. Nilpotent and Solvable Groups
 * 8. Normal and Subnormal Series


 * Chapter $\text {III}$: 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 $\text {IV}$: 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 $\text {V}$: Fields and Galois Theory
 * 1. Field Extensions
 * Appendix: Ruler and Compass Construction
 * 2. The Fundamental Theorem
 * Appendix: Symmetric Rational Functions
 * 3. Splitting Fields
 * 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 $\text {VI}$: The Structure of Fields
 * 1. Transcendence Bases
 * 2. Linear Disjointness and Separability


 * Chapter $\text {VII}$: 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 $\text {VIII}$: 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 Nullstellensatz


 * Chapter $\text {IX}$: 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 $\text {X}$: Categories
 * 1. Functors and Natural Transformations
 * 2. Adjoint Functors
 * 3. Morphisms


 * List of Symbols


 * Bibliography


 * Index