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 Evariste 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 Polya 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 Erdoes 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