Book:Thomas W. Hungerford/Algebra

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Thomas W. Hungerford: Algebra

Published $1974$, Springer-Verlag New York, Inc.

ISBN 0-387-90518-9.


Subject Matter


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