Contemporary Abstract Algebra / Edition 5

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Joseph Gallian is a well-known active researcher and award-winning teacher. His Contemporary Abstract Algebra, 6/e, includes challenging topics in abstract algebra as well as numerous figures, tables, photographs, charts, biographies, computer exercises, and suggested readings that give the subject a current feel and makes the content interesting and relevant for students.

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Product Details

  • ISBN-13: 9780618122141
  • Publisher: Houghton Mifflin Company College Division
  • Publication date: 7/1/1901
  • Edition description: Older Edition
  • Edition number: 5
  • Pages: 576
  • Product dimensions: 6.50 (w) x 9.12 (h) x 0.87 (d)

Table of Contents


Note: Each chapter concludes with Exercises.

  • I. Integers and Equivalence Relations
  • Preliminaries
    Properties of Integers
    Modular Arithmetic
    Mathematical Induction
    Equivalence Relations
    Functions (Mappings)
    Computer Exercises
    • II. Groups
    • 1. Introduction to Groups
      Symmetries of a Square
      The Dihedral Groups
      Biography of Neils Abel
      • 2. Groups
        Definition and Examples of Groups
        Elementary Properties of Groups
        Historical Note
        Computer Exercises
        • 3. Finite Groups; Subgroups
          Terminology and Notation
          Subgroup Tests
          Examples of Subgroups
          Computer Exercises
          • 4. Cyclic Groups
            Properties of Cyclic Groups
            Classification of Subgroups of Cyclic Groups
            Computer Exercises
            Biography of J. J. Sylvester
            • Supplementary Exercises for Chapters 1–4
            • 5. Permutation Groups
              Definition and Notation
              Cycle Notation
              Properties of Permutations
              A Check-Digit Scheme Based on D5
              Computer Exercises
              Biography of Augustin Cauchy
              • 6. Isomorphisms
                Definition and Examples
                Cayley's Theorem
                Properties of Isomorphisms
                Biography of Arthur Cayley
                • 7. Cosets and Lagrange's Theorem
                  Properties of Cosets
                  Lagrange's Theorem and Consequences
                  An Application of Cosets to Permutation Groups
                  The Rotation Group of a Cube and a Soccer Ball
                  Computer Exercises
                  Biography of Joseph Lagrange
                  • 8. External Direct Products
                    Definition and Examples
                    Properties of External Direct Products
                    The Group of Units Modulo n as an External DirectProduct
                    Computer Exercises
                    Biography of Leonard Adleman
                    • Supplementary Exercises for Chapters 5–8
                    • 9. Normal Subgroups and Factor Groups
                      Normal Subgroups
                      Factor Groups
                      Applications of Factor Groups
                      Internal Direct Products
                      Biography of Évariste Galois
                      • 10. Group Homomorphisms
                        Definition and Examples
                        Properties of Homomorphisms
                        The First Isomorphism Theorem
                        Computer Exercises
                        Biography of Camille Jordan
                        • 11. Fundamental Theorem of Finite Abelian Groups
                          The Fundamental Theorem
                          The Isomorphism Classes of Abelian Groups
                          Proof of the Fundamental Theorem
                          Computer Exercises
                          • Supplementary Exercises for Chapters 9–11
                          • III. Rings
                          • 12. Introduction to Rings
                            Motivation and Definition
                            Examples of Rings
                            Properties of Rings
                            Computer Exercises
                            Biography of I. N. Herstein
                            • 13. Integral Domains
                              Definition and Examples
                              Characteristic of a Ring
                              Computer Exercises
                              Biography of Nathan Jacobson
                              • 14. Ideals and Factor Rings
                                Factor Rings
                                Prime Ideals and Maximal Ideals
                                Biography of Richard Dedekind
                                Biography of Emmy Noether
                                • Supplementary Exercises for Chapters 12–14
                                • 15. Ring Homomorphisms
                                  Definition and Examples
                                  Properties of Ring Homomorphisms
                                  The Field of Quotients
                                  • 16. Polynomial Rings
                                    Notation and Terminology
                                    The Division Algorithm and Consequences
                                    Biography of Saunders Mac Lane
                                    • 17. Factorization of Polynomials
                                      Reducibility Tests
                                      Irreducibility Tests
                                      Unique Factorization in Z [x]
                                      Weird Dice: An Application of Unique Factorization
                                      Computer Exercises
                                      • 18. Divisibility in Integral Domains
                                        Irreducibles, Primes
                                        Historical Discussion of Fermat's Last Theorem
                                        Unique Factorization Domains
                                        Euclidean Domains
                                        Biography of Sophie Germain
                                        Biography of Andrew Wiles
                                        • Supplementary Exercises for Chapters 15–18
                                        • IV. Fields
                                        • 19. Vector Spaces
                                          Definition and Examples
                                          Linear Independence
                                          Biography of Emil Artin
                                          Biography of Olga Taussky-Todd
                                          • 20. Extension Fields
                                            The Fundamental Theorem of Field Theory
                                            Splitting Fields
                                            Zeros of an Irreducible Polynomial
                                            Biography of Leopold Kronecker
                                            • 21. Algebraic Extensions
                                              Characterization of Extensions
                                              Finite Extensions
                                              Properties of Algebraic Extensions
                                              Biography of Irving Kaplansky
                                              • 22. Finite Fields
                                                Classification of Finite Fields
                                                Structure of Finite Fields
                                                Subfields of a Finite Field
                                                Computer Exercises
                                                Biography of L. E. Dickson
                                                • 23. Geometric Constructions
                                                  Historical Discussion of Geometric Constructions
                                                  Constructible Numbers
                                                  Angle-Trisectors and Circle-Squarers
                                                  • Supplementary Exercises for Chapters 19–23
                                                  • V. Special Topics
                                                  • 24. Sylow Theorems
                                                    Conjugacy Classes
                                                    The Class Equation
                                                    The Probability That Two Elements Commute
                                                    The Sylow Theorems
                                                    Applications of Sylow Theorems
                                                    Biography of Ludvig Sylow
                                                    • 25. Finite Simple Groups
                                                      Historical Background
                                                      Nonsimplicity Tests
                                                      The Simplicity of A5
                                                      The Fields Medal
                                                      The Cole Prize
                                                      Computer Exercises
                                                      Biography of Michael Aschbacher
                                                      Biography of Daniel Gorenstein
                                                      Biography of John Thompson
                                                      • 26. Generators and Relations
                                                        Definitions and Notation
                                                        Free Group
                                                        Generators and Relations
                                                        Classification of Groups of Order up to 15
                                                        Characterization of Dihedral Groups
                                                        Realizing the Dihedral Groups with Mirrors
                                                        Biography of Marshall Hall, Jr.
                                                        • 27. Symmetry Groups
                                                          Classification of Finite Plane Symmetry Groups
                                                          Classification of Finite Group of Rotations in R3
                                                          • 28. Frieze Groups and Crystallographic Groups
                                                            The Frieze Groups
                                                            The Crystallographic Groups
                                                            Identification of Plane Periodic Patterns
                                                            Biography of M. C. Escher
                                                            Biography of George Pólya
                                                            Biography of John H. Conway
                                                            • 29. Symmetry and Counting
                                                              Burnside's Theorem
                                                              Group Action
                                                              Biography of William Burnside
                                                              • 30. Cayley Digraphs of Groups
                                                                The Cayley Digraph of a Group
                                                                Hamiltonian Circuits and Paths
                                                                Some Applications
                                                                Biography of William Rowan Hamilton
                                                                Biography of Paul Erdös
                                                                • 31. Introduction to Algebraic Coding Theory
                                                                  Linear Codes
                                                                  Parity-Check Matrix Decoding
                                                                  Coset Decoding
                                                                  Historical Note: Reed-Solomon Codes
                                                                  Biography of Richard W. Hamming
                                                                  Biography of Jessie MacWilliams
                                                                  Biography of Vera Pless
                                                                  • 32. An Introduction to Galois Theory
                                                                    Fundamental Theorem of Galois Theory
                                                                    Solvability of Polynomials by Radicals
                                                                    Insolvability of a Quintic
                                                                    Biography of Philip Hall
                                                                    • 33. Cyclotomic Extensions
                                                                      Cyclotomic Polynomials
                                                                      The Constructible Regular n-gons
                                                                      Computer Exercise
                                                                      Biography of Carl Friedrich Gauss
                                                                      Biography of Manjul Bhargava
                                                                      • Supplementary Exercises for Chapters 24–33
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  • Anonymous

    Posted December 21, 2004

    The best book for learning abstract algebra

    This book has everything a math textbook should have. The examples are apt and abundant. The proofs are well-phrased, with just enough detail but not too much. There's a smattering of interesting applications (and toys). And the exercises are downright brilliant--never too hard, but never easy. But there's even more to this book: Gallian's vision of algebra as a centuries-spanning collaborative effort shines from every page. I have never encountered such a smart book or one better suited to self-education. I unreservedly recommend it.

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  • Anonymous

    Posted February 12, 2004

    A great book

    This is a great coursebook on Abstract Algebra. The level is suitable for students of undergraduate to advanced-undergraduate caliber. Abstract Algebra is covered in sections from Basic Concepts to Groups to Rings and finally Fields with an optional fifth section on Advanced Topics including Galois Theory and the Sylow Theorem.

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