Review of Book”Contemporary Abstract Algebra by Joseph Gallian”

Contemporary Abstract Algebra by Joseph Gallian

Abstract Reviews–

abs Review of Book''Contemporary Abstract Algebra by Joseph Gallian''Joseph Gallian is a well-known active researcher and award-winning teacher. His Contemporary Abstract Algebra 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.

There are very interesting changes for the ninth edition include new exercises, new examples, new biographies, new quotes, new applications, and a freshening of the historical notes and biographies from the 8th edition. These changes accentuate and enhance the hallmark features that have made previous editions of the book a comprehensive, lively, and engaging introduction to the subject:


  • Extensive coverage of groups, rings, and fields, plus a variety of non-traditional special topics
  • A good mixture of more nearly 1700 computational and theoretical exercises appearing in each chapter that synthesizes concepts from multiple chapters
  • Back-of-the-book skeleton solutions and hints to the odd-numbered exercises
  • Worked-out examples– totaling more than 300–ranging from routine computations to quite challenging
  • Computer exercises that utilize interactive software available on my website that stress guessing and making conjectures
  • A large number of applications from scientific and computing fields, as well as from everyday life
  • Numerous historical notes and biographies that spotlight the people and events behind the mathematics
  • Motivational and humorous quotations.
  • More than 275 figures, photographs, tables, and reproductions of currency that honor mathematicians
  • Annotated suggested readings for interesting further exploration of topics.
    • True/false questions with comments
    • Flash cards
    • Essays on learning abstract algebra, doing proofs, and reasons whyabstract algebra is a valuable subject to learn
    • Links to abstract algebra-related websites and software packages andmuch, much more.

    This book is OK as far as presenting abstract algebra in the usual way to undergraduates. Competent explanations of the basics of groups, rings, and fields. Numerous easy exercises, which is fine, although it might be nice if there were more challenging ones too.

    → This book provides a solid introduction to the traditional topics in abstract algebra while conveying to students that it is a contemporary subject used daily by working mathematicians, computer scientists, physicists, and chemists.

    → The text includes numerous figures, tables, photographs, charts, biographies, computer exercises, and suggested readings giving the subject a current feel which makes the content interesting and relevant for students.


    Table Contents of This Excellent Book


     

  • I. Integers and Equivalence Relationsabs Review of Book''Contemporary Abstract Algebra by Joseph Gallian''
  • 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
    Motivation
    Definition and Examples
    Cayley’s Theorem
    Properties of Isomorphisms
    Automorphisms
    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 Direct Product
    Applications
    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
    Subrings
    Computer Exercises
    Biography of I. N. Herstein
  • 13. Integral Domains
    Definition and Examples
    Fields
    Characteristic of a Ring
    Computer Exercises
    Biography of Nathan Jacobson
  • 14. Ideals and Factor Rings
    Ideals
    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
    Subspaces
    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
    Motivation
    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
    Isometries
    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
    Motivation
    Burnside’s Theorem
    Applications
    Group Action
    Biography of William Burnside
  • 30. Cayley Digraphs of Groups
    Motivation
    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
    Motivation
    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
    Motivation
    Cyclotomic Polynomials
    The Constructible Regular n-gons
    Computer Exercise
    Biography of Carl Friedrich Gauss
    Biography of Manjul Bhargava
  • Supplementary Exercises for Chapters 24-33

abs Review of Book''Contemporary Abstract Algebra by Joseph Gallian''abs Review of Book''Contemporary Abstract Algebra by Joseph Gallian''

There are some excellent snapshots from this book.




abs Review of Book''Contemporary Abstract Algebra by Joseph Gallian''

abs Review of Book''Contemporary Abstract Algebra by Joseph Gallian''




abs Review of Book''Contemporary Abstract Algebra by Joseph Gallian''

abs Review of Book''Contemporary Abstract Algebra by Joseph Gallian''