Preface | p. xi |

Integers and Equivalence Relations | p. 1 |

Preliminaries | p. 3 |

Properties of Integers | p. 3 |

Madular Arithmetic | p. 7 |

Mathematical Induction | p. 12 |

Equivalence Relations | p. 15 |

Functions (Mappings) | p. 18 |

Exercises | p. 21 |

Computer Exercises | p. 25 |

Groups | p. 27 |

Introduction to Groups | p. 29 |

Symmetries of a Square | p. 29 |

The Dihedral Groups | p. 32 |

Exercises | p. 35 |

Biography of Niels Abel | p. 39 |

Groups | p. 40 |

Definition and Examples of Groups o40 | |

Elementary Properties of Groups | p. 48 |

Historical Note | p. 51 |

Exercises | p. 52 |

Computer Exercises | p. 55 |

Finite Groups; Subgroups | p. 57 |

Terminology and Notation | p. 57 |

Subgroup Tests | p. 58 |

Examples of Subgroups | p. 61 |

Exercises | p. 64 |

Computer Exercises | p. 70 |

Cyclic Groups | p. 72 |

Properties of Cycle Groups | p. 72 |

Classification of Subgroups of Cyclic Groups | p. 77 |

Exercises | p. 81 |

Computer Exercises | p. 86 |

Biography of J. J. Sylvester | p. 89 |

Supplementary Exercises for Chapters1-4 | p. 91 |

Permutation Groups | p. 95 |

Difinition and Notation | p. 95 |

Cycle Nation | p. 98 |

Properties of Permutations | p. 100 |

A Check Digit Scheme Based on D5 | p. 110 |

Exercises | p. 113 |

Computer Exercises | p. 118 |

Biography of Augustin Cauchy | p. 121 |

Isomorphisms | p. 122 |

Motivation | p. 122 |

Dfinition and Examples | p. 122 |

CayleyÆs Theorem | p. 126 |

Properties of Isomorphisms | p. 128 |

Automorphisms | p. 129 |

Exercises | p. 133 |

Computer Exercise | p. 136 |

Biography of Arthur Cayley | p. 137 |

Cosets and LagrangeÆs Theorem | p. 138 |

Properties of Cosets | p. 138 |

LagrangeÆs Theorem and Consequences | p. 141 |

An Application of Cosets of Permutation Groups | p. 145 |

The Rotation Group of a Cube and a Soccer Ball | p. 146 |

Exercises | p. 149 |

Computer Exercise | p. 153 |

Biography of Joseph Lagrange | p. 154 |

External Direct Products | p. 155 |

Definition and Examples | p. 155 |

Properties of External Direct Products | p. 156 |

The Group of Units Modulo n as an External Direct Products | p. 159 |

Applications | p. 161 |

Exercises | p. 167 |

Computer Exercises | p. 170 |

Biorgaphy of Leonard Adleman | p. 173 |

Supplementary Exercises for Chapters 5-8 | p. 174 |

Normal Subgroups and Factor Groups | p. 178 |

Normal Subgroups | p. 178 |

Factor Groups | p. 180 |

Applicatons of Factor Groups | p. 185 |

Internal Direct Products | p. 188 |

Exercises | p. 193 |

Biography of Evariste Galois | p. 199 |

Group Homomorphisms | p. 200 |

Difinition and Examples | p. 200 |

Properties Of Homomorphisms | p. 202 |

The First Isomorphism Theorem | p. 206 |

Exercises | p. 211 |

Computer Exercise | p. 216 |

Biography of Camille Jordan | p. 217 |

Fundamental Theorem of Finite Abelian Groups | p. 218 |

The Fundamental Theorem | p. 218 |

The Isomorphism Classes of Abelian Groups | p. 218 |

Proof of the Fundamental Theorem | p. 223 |

Exercises | p. 226 |

Computer Exercises | p. 228 |

Supplementary Exercises for Chapter 9-11 | p. 230 |

Rings | p. 235 |

Introduction to Rings | p. 237 |

Motivation and Definition | p. 237 |

Examples of Rings | p. 238 |

Properties of Rings | p. 239 |

Subrings | p. 240 |

Exercises | p. 242 |

Computer Exercises | p. 245 |

Biography of I. N. Herstein | p. 248 |

Integral Domains | p. 249 |

Definition and Examples | p. 249 |

Fields | p. 250 |

Characteristic of a Ring | p. 225 |

Exercises | p. 255 |

Computer Exercises | p. 259 |

Biography of Nathan Jacobson | p. 261 |

Ideals and Factor Rings | p. 262 |

Ideals | p. 262 |

Factor Rings | p. 263 |

Prime Ideals and Maximal Ideals | p. 267 |

Exercises | p. 269 |

Computer Exercises | p. 273 |

Biography of Richard Dedekind | p. 274 |

Biography of Emmy Noether | p. 275 |

Supplementary Exercises for Chapters 12-14 | p. 276 |

Ring Homomorphisms | p. 280 |

Definition and Example | p. 280 |

Properties of Ring Homomorphisms | p. 283 |

The Field of Quotients | p. 285 |

Exercises | p. 287 |

Polynomial Rings | p. 293 |

Notation and Terminology | p. 293 |

The Division Algorithm and Consequences | p. 296 |

Exercises | p. 300 |

Biography of Sounders Mac Lane | p. 304 |

Factorization of Polynomials | p. 305 |

Reducibility Tests | p. 305 |

Irreducibility Tests | p. 308 |

Unique Factorization in Z[x] | p. 313 |

Weird Dice: An Application of Unique Factorization | p. 314 |

Exercises | p. 316 |

Computer Exercises | p. 319 |

Biography of Serge Lang | p. 321 |

Divisibility in Integral Domains | p. 322 |

Irreducibles, Primes | p. 322 |

Historical Discussion of FermatÆs Last Theorem | p. 325 |

Unique Factorization Domains | p. 328 |

Euclidean Domains | p. 331 |

Exercises | p. 335 |

Comupter Exercise | p. 337 |

Biography of Sophie Germain | p. 339 |

Biography of Andrew Wiles | p. 340 |

Supplementary Exercises for Chapters 15-18 | p. 341 |

Fields | p. 343 |

Vector Spaces | p. 345 |

Definition and Examples | p. 345 |

Subspaces | |

Linear Independence | p. 347 |

Exercises | p. 349 |

Biography of Emil Artin | p. 352 |

Biography of Olga Taussky-Todd | p. 353 |

Extension Fields | p. 354 |

The Fundamental Theorem of Field theory | p. 354 |

Splitting Fields | p. 356 |

Zeros of an Irreducible Polynomial | p. 362 |

Exercises | p. 366 |

Biography of Leopold Kronecker | p. 369 |

Algebraci Extensions | p. 370 |

Characterization of Extensions | p. 370 |

Finite Extensions | p. 372 |

Properties of Algebraic Extensions | p. 376 |

Exercises | p. 378 |

Biography of Irving Kaplansky | p. 381 |

Finite Fields | p. 382 |

Classification of Finite Fields | p. 382 |

Struction of Finite Fields | p. 383 |

Subfields of a Finite Field | p. 387 |

Exercises | p. 389 |

Computer Exercises | p. 391 |

Biography of L. E. Dickson | p. 392 |

Geometric Constructions | p. 393 |

Historical Discussion of Geometric Constructions | p. 393 |

Constructible Numbers | p. 394 |

Angle-Trisectors and Circle-Squarers | p. 396 |

Exercises | p. 396 |

Supplementary Exercises for Chapters | p. 19-23 |

Special Topics | p. 401 |

Sylow Theorems | p. 403 |

Conjugacy Classes | p. 403 |

The Class Equation | p. 404 |

The Probability That Two Elements Commute | p. 405 |

The Sylow Theorems | p. 406 |

Applications of Sylow Theorems | p. 411 |

Exercises | p. 414 |

Computer Exercise | p. 418 |

Biography of Ludwig Sylow | p. 419 |

Finite Simple Groups | p. 420 |

Historical Background | p. 420 |

Nonsimplicity Tests | p. 245 |

The Simplicity of A5 | p. 429 |

The Fields Medal | p. 430 |

The Cole Prize | p. 430 |

Execises | p. 431 |

Computer Exercises | p. 432 |

Biography of Michael Aschbacher | p. 434 |

Biography of Daniel Gorenstein | p. 435 |

Biography of John Thompson | p. 436 |

Generators and Relations | p. 437 |

Motivation | p. 437 |

Definitions and Notation | p. 438 |

Free Group | p. 439 |

Generators and Relations | p. 440 |

Classification of Groups of Order Up to 15 | p. 444 |

Characterization of Dihedral Group | p. 446 |

Realizing the Dihedral Groups with Mirrors | p. 447 |

Exercises | p. 449 |

Biography of Marshall Hall, Jr. | p. 452 |

Symmetry Groups | p. 453 |

Isometries | p. 453 |

Classification of Finite Plane Symmetry Group | p. 455 |

Classification of Finite Groups of Rotations in R3 | p. 456 |

Exercises | p. 458 |

Frieze Groups and Crystallographic Groups | p. 461 |

The Frieze Groups | p. 461 |

The Crystallographic Groups | p. 467 |

Identification of Plane Periodic Patterns | p. 473 |

Exercises | p. 479 |

Biography of M. C. Escher | p. 484 |

Biography of George Polya | p. 485 |

Biography of John H. Conway | p. 486 |

Symmetry and Counting | p. 487 |

Motivation | p. 487 |

BurnsideÆs Theorem | p. 488 |

Applications | p. 490 |

Group Action | p. 493 |

Exercises | p. 494 |

Biography of William Burnside | p. 497 |

Cayley Digraphs of Groups | p. 498 |

Motivaton | p. 498 |

The Cayley Digraph of a Group | p. 498 |

Hamiltonian Circuits and Paths | p. 502 |

Some Apllications | p. 508 |

Exercises | p. 511 |

Biography of William Rowan Hamilton | p. 516 |

Biography of Paul Erdos | p. 517 |

Indtoduction to Algebraic Coding Theory | p. 518 |

Motivation | p. 518 |

Liner Codes | p. 523 |

Parity-Check Matrix Decoding | p. 528 |

Coset Decoding | p. 531 |

Hestorical Note: The Ubiquitous Reed-Solomon Codes | p. 535 |

Exercises | p. 537 |

Biography of Richard W. Hamming | p. 542 |

Biography of Jessie Mac Williams | p. 543 |

Biography of Vera Pless | p. 544 |

An Introduction to Galois Theory | p. 545 |

Fundamental Theorem of Galois Theory | p. 545 |

Solvability of Polynomials by Radicals | p. 552 |

Insolvability of a Quintic | p. 556 |

Exercises | p. 557 |

Biography of Philip Hall | p. 560 |

Cyclotomic Extensions | p. 561 |

Motivation | p. 561 |

Cyclotomic Polynomials | p. 562 |

The Constructible Regular n-Gons | p. 566 |

Exercises | p. 568 |

Computer Exercis | p. 569 |

Biography of Carl Friedrich Gauss | p. 570 |

Biography of Manjul Bhargava | p. 571 |

Supplementary Exercises for Chapters 24-33 | p. 572 |

Selected Answers | p. A1 |

Text Credits | p. A40 |

Photo Credits | p. A42 |

Index of Mathematicians | p. A43 |

Index of Terms | p. A45 |

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