I. Integers and Equivalence Relations
0. 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 Exercise
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
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 Evariste Galois

10. Group Homomorphisms

Definition and Examples
Properties of Homomorphisms
The First Isomorphism Theorem
Biography Camille Jordan

11. Fundamental Theorem of Finite Abelian Groups

The Fundamental Theorem
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
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
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 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 William Rowan Hamilton
Biography 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 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 Philip Hall

33. Cyclotomic Extensions

Motivation
Cyclotomic Polynomials
The Constructible Regular n-gons
Computer Exercise
Biography Carl Friedrich Gauss
Supplementary Exercises Ch. 24-33