Book:Thomas W. Hungerford/Algebra

Thomas W. Hungerford: Algebra

Published $\text {1974}$, Springer-Verlag New York, Inc.

ISBN 0-387-90518-9

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