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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 ï¿½variste 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 Pï¿½lya | |

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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 | |