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
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

About the Author

Joseph Gallian earned his PhD from Notre Dame. In addition to receiving numerous awards for his teaching and exposition, he served, first, as the Second Vice President, and, then, as the President of the MAA. He has served on 40 national committees, chairing ten of them. He has published over 100 articles and authored six books. Numerous articles about his work have appeared in the national news outlets, including the New York Times, the Washington Post, the Boston Globe, and Newsweek, among many others.