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| Preface | |
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| Integers and Equivalence Relations | |
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| Preliminaries | |
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| Properties of Integers | |
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| Modular Arithmetic | |
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| Complex Numbers | |
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| Mathematical Induction | |
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| Equivalence Relations | |
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| Functions (Mappings) | |
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| Exercises | |
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| Groups | |
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| Introduction to Groups | |
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| Symmetries of a Square | |
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| The Dihedral Groups | |
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| Exercises | |
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| Biography of Niels Abel | |
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| Groups | |
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| Definition and Examples of Groups | |
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| Elementary Properties of Groups | |
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| Historical Note | |
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| Exercises | |
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| Finite Groups; Subgroups | |
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| Terminology and Notation | |
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| Subgroup Tests | |
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| Examples of Subgroups | |
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| Exercises | |
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| Cyclic Groups | |
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| Properties of Cyclic Groups | |
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| Classification of Subgroups of Cyclic Groups | |
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| Exercises | |
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| Biography of James Joseph Sylvester | |
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| Supplementary Exercises for Chapters 1-4 | |
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| Permutation Groups | |
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| Definition and Notation | |
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| Cycle Notation | |
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| Properties of Permutations | |
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| A Check-Digit Scheme Based on D<sub>5</sub> | |
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| Exercises | |
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| Biography of Augustin Cauchy | |
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| Isomorphisms | |
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| Motivation | |
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| Definition and Examples | |
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| Cayley's Theorem | |
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| Properties of Isomorphisms | |
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| Automorphisms | |
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| Exercises | |
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| Biography of Arthur Cayley | |
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| Cosets and Lagrange's Theorem | |
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| Properties of Cosets | |
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| Lagrange's Theorem and Consequences | |
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| An Application of Cosets to Permutation Groups | |
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| The Rotation Group of a Cube and a Soccer Ball | |
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| An Application of Cosets to the Rubik's Cube | |
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| Exercises | |
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| Biography of Joseph Lagrange | |
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| External Direct Products | |
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| Definition and Examples | |
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| Properties of External Direct Products | |
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| The Group of Units Modulo n as an External Direct Product | |
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| Applications | |
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| Exercises | |
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| Biography of Leonard Adleman | |
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| Supplementary Exercises for Chapters 5-8 | |
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| Normal Subgroups and Factor Groups | |
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| Normal Subgroups | |
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| Factor Groups | |
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| Applications of Factor Groups | |
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| Internal Direct Products | |
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| Exercises | |
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| Biography of �variste Galois | |
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| Group Homomorphisms | |
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| Definition and Examples | |
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| Properties of Homomorphisms | |
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| The First Isomorphism Theorem | |
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| Exercises | |
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| Biography of Camille Jordan | |
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| Fundamental Theorem of Finite Abelian Groups | |
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| The Fundamental Theorem | |
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| The Isomorphism Classes of Abelian Groups | |
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| Proof of the Fundamental Theorem | |
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| Exercises | |
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| Supplementary Exercises for Chapters 9-11 | |
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| Rings | |
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| Introduction to Rings | |
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| Motivation and Definition | |
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| Examples of Rings | |
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| Properties of Rings | |
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| Subrings | |
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| Exercises | |
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| Biography of I. N. Herstein | |
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| Integral Domains | |
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| Definition and Examples | |
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| Fields | |
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| Characteristic of a Ring | |
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| Exercises | |
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| Biography of Nathan Jacobson | |
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| Ideals and Factor Rings | |
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| Ideals | |
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| Factor Rings | |
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| Prime Ideals and Maximal Ideals | |
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| Exercises | |
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| Biography of Richard Dedekind | |
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| Biography of Emmy Noether | |
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| Supplementary Exercises for Chapters 12-14 | |
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| Ring Homomorphisms | |
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| Definition and Examples | |
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| Properties of Ring Homomorphisms | |
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| The Field of Quotients | |
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| Exercises | |
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| Polynomial Rings | |
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| Notation and Terminology | |
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| The Division Algorithm and Consequences | |
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| Exercises | |
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| Biography of Saunders Mac Lane | |
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| Factorization of Polynomials | |
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| Reducibility Tests | |
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| Irreducibility Tests | |
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| Unique Factorization in Z[x] | |
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| Weird Dice: An Application of Unique Factorization | |
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| Exercises | |
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| Biography of Serge Lang | |
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| Divisibility in Integral Domains | |
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| Irreducibles, Primes | |
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| Historical Discussion of Fermat's Last Theorem | |
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| Unique Factorization Domains | |
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| Euclidean Domains | |
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| Exercises | |
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| Biography of Sophie Germain | |
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| Biography of Andrew Wiles | |
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| Supplementary Exercises for Chapters 15-18 | |
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| Fields | |
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| Vector Spaces | |
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| Definition and Examples | |
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| Subspaces | |
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| Linear Independence | |
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| Exercises | |
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| Biography of Emil Artin | |
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| Biography of Olga Taussky-Todd | |
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| Extension Fields | |
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| The Fundamental Theorem of Field Theory | |
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| Splitting Fields | |
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| Zeros of an Irreducible Polynomial | |
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| Exercises | |
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| Biography of Leopold Kronecker | |
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| Algebraic Extensions | |
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| Characterization of Extensions | |
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| Finite Extensions | |
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| Properties of Algebraic Extensions | |
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| Exercises | |
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| Biography of Irving Kaplansky | |
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| Finite Fields | |
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| Classification of Finite Fields | |
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| Structure of Finite Fields | |
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| Subfields of a Finite Field | |
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| Exercises | |
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| Biography of L. E. Dickson | |
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| Geometric Constructions | |
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| Historical Discussion of Geometric Constructions | |
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| Constructible Numbers | |
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| Angle-Trisectors and Circle-Squarers | |
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| Exercises | |
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| Supplementary Exercises for Chapters 19-23 | |
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| Special Topics | |
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| Sylow Theorems | |
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| Conjugacy Classes | |
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| The Class Equation | |
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| The Probability That Two Elements Commute | |
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| The Sylow Theorems | |
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| Applications of Sylow Theorems | |
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| Exercises | |
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| Biography of Ludwig Sylow | |
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| Finite Simple Groups | |
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| Historical Background | |
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| Nonsimplicity Tests | |
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| The Simplicity of A<sub>5</sub> | |
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| The Fields Medal | |
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| The Cole Prize | |
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| Exercises | |
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| Biography of Michael Aschbacher | |
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| Biography of Daniel Gorenstein | |
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| Biography of John Thompson | |
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| Generators and Relations | |
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| Motivation | |
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| Definitions and Notation | |
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| Free Group | |
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| Generators and Relations | |
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| Classification of Groups of Order Up to 15 | |
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| Characterization of Dihedral Groups | |
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| Realizing the Dihedral Groups with Mirrors | |
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| Exercises | |
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| Biography of Marshall Hall, Jr. | |
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| Symmetry Groups | |
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| Isometries | |
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| Classification of Finite Plane Symmetry Groups | |
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| Classification of Finite Groups of Rotations in R<sup>3</sup> | |
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| Exercises | |
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| Frieze Groups and Grystallographic Groups | |
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| The Frieze Groups | |
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| The Crystallographic Groups | |
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| Identification of Plane Periodic Patterns | |
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| Exercises | |
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| Biography of M. C. Escher | |
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| Biography of George P�lya | |
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| Biography of John H. Conway | |
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| Symmetry and Counting | |
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| Motivation | |
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| Burnside's Theorem | |
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| Applications | |
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| Group Action | |
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| Exercises | |
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| Biography of William Burnside | |
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| Cayley Digraphs of Groups | |
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| Motivation | |
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| The Cayley Digraph of a Group | |
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| Hamiltonian Circuits and Paths | |
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| Some Applications | |
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| Exercises | |
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| Biography of William Rowan Hamilton | |
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| Biography of Paul Erdos | |
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| Introduction to Algebraic Coding Theory | |
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| Motivation | |
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| Linear Codes | |
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| Parity-Check Matrix Decoding | |
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| Coset Decoding | |
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| Historical Note: The Ubiquitous Reed-Solomon Codes | |
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| Exercises | |
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| Biography of Richard W. Hamming | |
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| Biography of Jessie MacWilliams | |
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| Biography of Vera Pless | |
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| An Introduction to Galois Theory | |
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| Fundamental Theorem of Galois Theory | |
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| Solvability of Polynomials by Radicals | |
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| Insolvability of a Quintic | |
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| Exercises | |
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| Biography of Philip Hall | |
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| Cyclotomic Extensions | |
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| Motivation | |
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| Cyclotomic Polynomials | |
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| The Constructible Regular n-gons | |
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| Exercises | |
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| Biography of Carl Friedrich Gauss | |
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| Biography of Manjul Bhargava | |
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| Supplementary Exercises for Chapters 24-33 | |
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| Selected Answers | |
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| Index of Mathematicians | |
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| Index of Terms | |