Table of Contents

# Contemporary Abstract Algebra by *Joseph Gallian*

## Abstract Reviews–

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 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
*D*5 - 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
*A*5 - 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
**R**3

- 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**

# There are some excellent snapshots from this book.