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Algebra

Thomas W. Hungerford
Publisher: 
Springer Verlag
Publication Date: 
2003
Number of Pages: 
528
Format: 
Hardcover
Series: 
Graduate Texts in Mathematics 73
Price: 
54.95
ISBN: 
0387905189
Category: 
Textbook
BLL Rating: 
BLL*

The Basic Library List Committee recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
Allen Stenger
, on
09/24/2014
]

This is a text for a first-year graduate course in abstract algebra. It covers all the standard topics and has more than enough material for a year course. It is self-contained, but assumes the student has already had an undergraduate modern algebra course. The present volume is a 2003 corrected reprint of the 1974 Holt, Rinehart and Winston volume.

A comparable book is Dummit & Foote’s Abstract Algebra. Despite being thirty years newer and having about twice as many pages, this book has generally the same coverage and approach. Dummit and Foote do have some newer topics such as Gröbner bases. Their greater length comes primarily from a larger number of worked examples (that are also worked out in more detail), more exercises (both books are well-supplied with exercises), and from being more chatty (Hungerford follows the traditional Theorem-Proof exposition, with occasional road signs to tell us where we are going).

Hungerford’s newer book Abstract Algebra: An Introduction is a very different book. It is a good undergraduate introduction to the present book, but only overlaps it by about 50%. The newer book has a much larger number of applications, many of them recent.

Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis.

  • 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 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 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
    • 1. Transcendence Bases
    • 2. Linear Disjointness and Separability
  • Chapter 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 VIII: Commutative Rings and Modules
    • 1. Chain Conditions
    • 2. Prime and Primary Ideals
    • 3. Primary Decomposition
    • 4. Noetherian Rings and Modules
    • S. Ring Extensions
    • 6. Dedekind Domains
    • 7. The Hilbert Nullstellensatz
  • 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
Tags: 
Abstract Algebra
Algebra