Preface | p. xi |

Integers and Equivalence Relations | p. 1 |

Preliminaries | p. 3 |

Properties of Integers | p. 3 |

Modular Arithmetic | p. 6 |

Complex Numbers | p. 13 |

Mathematical Induction | p. 14 |

Equivalence Relations | p. 17 |

Functions (Mappings) | p. 20 |

Exercises | p. 23 |

Groups | p. 29 |

Introduction to Groups | p. 31 |

Symmetries of a Square | p. 31 |

The Dihedral Groups | p. 34 |

Exercises | p. 37 |

Biography of Niels Abel | p. 41 |

Groups | p. 42 |

Definition and Examples of Groups | p. 42 |

Elementary Properties of Groups | p. 50 |

Historical Note | p. 53 |

Exercises | p. 54 |

Finite Groups; Subgroups | p. 60 |

Terminology and Notation | p. 60 |

Subgroup Tests | p. 61 |

Examples of Subgroups | p. 65 |

Exercises | p. 68 |

Cyclic Groups | p. 77 |

Properties of Cyclic Groups | p. 77 |

Classification of Subgroups of Cyclic Groups | p. 82 |

Exercises | p. 87 |

Biography of James Joseph Sylvester | p. 93 |

Supplementary Exercises for Chapters 1-4 | p. 95 |

Permutation Groups | p. 99 |

Definition and Notation | p. 99 |

Cycle Notation | p. 102 |

Properties of Permutations | p. 104 |

A Check-Digit Scheme Based on D_{5} | p. 115 |

Exercises | p. 118 |

Biography of Augustin Cauchy | p. 126 |

Isomorphisms | p. 127 |

Motivation | p. 127 |

Definition and Examples | p. 127 |

Cayley's Theorem | p. 131 |

Properties of Isomorphisms | p. 133 |

Automorphisms | p. 134 |

Exercises | p. 138 |

Biography of Arthur Cayley | p. 143 |

Cosets and Lagrange's Theorem | p. 144 |

Properties of Cosets | p. 144 |

Lagrange's Theorem and Consequences | p. 147 |

An Application of Cosets to Permutation Groups | p. 151 |

The Rotation Group of a Cube and a Soccer Ball | p. 153 |

An Application of Cosets to the Rubik's Cube | p. 155 |

Exercises | p. 156 |

Biography of Joseph Lagrange | p. 161 |

External Direct Products | p. 162 |

Definition and Examples | p. 162 |

Properties of External Direct Products | p. 163 |

The Group of Units Modulo n as an External Direct Product | p. 166 |

Applications | p. 168 |

Exercises | p. 174 |

Biography of Leonard Adleman | p. 180 |

Supplementary Exercises for Chapters 5-8 | p. 181 |

Normal Subgroups and Factor Groups | p. 185 |

Normal Subgroups | p. 185 |

Factor Groups | p. 187 |

Applications of Factor Groups | p. 193 |

Internal Direct Products | p. 195 |

Exercises | p. 200 |

Biography of Évariste Galois | p. 207 |

Group Homomorphisms | p. 208 |

Definition and Examples | p. 208 |

Properties of Homomorphisms | p. 210 |

The First Isomorphism Theorem | p. 214 |

Exercises | p. 219 |

Biography of Camille Jordan | p. 225 |

Fundamental Theorem of Finite Abelian Groups | p. 226 |

The Fundamental Theorem | p. 226 |

The Isomorphism Classes of Abelian Groups | p. 226 |

Proof of the Fundamental Theorem | p. 231 |

Exercises | p. 234 |

Supplementary Exercises for Chapters 9-11 | p. 238 |

Rings | p. 243 |

Introduction to Rings | p. 245 |

Motivation and Definition | p. 245 |

Examples of Rings | p. 246 |

Properties of Rings | p. 247 |

Subrings | p. 248 |

Exercises | p. 250 |

Biography of I. N. Herstein | p. 254 |

Integral Domains | p. 255 |

Definition and Examples | p. 255 |

Fields | p. 256 |

Characteristic of a Ring | p. 258 |

Exercises | p. 261 |

Biography of Nathan Jacobson | p. 266 |

Ideals and Factor Rings | p. 267 |

Ideals | p. 267 |

Factor Rings | p. 268 |

Prime Ideals and Maximal Ideals | p. 272 |

Exercises | p. 274 |

Biography of Richard Dedekind | p. 279 |

Biography of Emmy Noether | p. 280 |

Supplementary Exercises for Chapters 12-14 | p. 281 |

Ring Homomorphisms | p. 285 |

Definition and Examples | p. 285 |

Properties of Ring Homomorphisms | p. 288 |

The Field of Quotients | p. 290 |

Exercises | p. 292 |

Polynomial Rings | p. 298 |

Notation and Terminology | p. 298 |

The Division Algorithm and Consequences | p. 301 |

Exercises | p. 305 |

Biography of Saunders Mac Lane | p. 310 |

Factorization of Polynomials | p. 311 |

Reducibility Tests | p. 311 |

Irreducibility Tests | p. 314 |

Unique Factorization in Z[x] | p. 319 |

Weird Dice: An Application of Unique Factorization | p. 320 |

Exercises | p. 322 |

Biography of Serge Lang | p. 327 |

Divisibility in Integral Domains | p. 328 |

Irreducibles, Primes | p. 328 |

Historical Discussion of Fermat's Last Theorem | p. 331 |

Unique Factorization Domains | p. 334 |

Euclidean Domains | p. 337 |

Exercises | p. 341 |

Biography of Sophie Germain | p. 345 |

Biography of Andrew Wiles | p. 346 |

Supplementary Exercises for Chapters 15-18 | p. 347 |

Fields | p. 349 |

Vector Spaces | p. 351 |

Definition and Examples | p. 351 |

Subspaces | p. 352 |

Linear Independence | p. 353 |

Exercises | p. 355 |

Biography of Emil Artin | p. 358 |

Biography of Olga Taussky-Todd | p. 359 |

Extension Fields | p. 360 |

The Fundamental Theorem of Field Theory | p. 360 |

Splitting Fields | p. 362 |

Zeros of an Irreducible Polynomial | p. 368 |

Exercises | p. 372 |

Biography of Leopold Kronecker | p. 375 |

Algebraic Extensions | p. 376 |

Characterization of Extensions | p. 376 |

Finite Extensions | p. 378 |

Properties of Algebraic Extensions | p. 382 |

Exercises | p. 384 |

Biography of Irving Kaplansky | p. 387 |

Finite Fields | p. 388 |

Classification of Finite Fields | p. 388 |

Structure of Finite Fields | p. 389 |

Subfields of a Finite Field | p. 393 |

Exercises | p. 395 |

Biography of L. E. Dickson | p. 398 |

Geometric Constructions | p. 399 |

Historical Discussion of Geometric Constructions | p. 399 |

Constructible Numbers | p. 400 |

Angle-Trisectors and Circle-Squarers | p. 402 |

Exercises | p. 402 |

Supplementary Exercises for Chapters 19-23 | p. 405 |

Special Topics | p. 407 |

Sylow Theorems | p. 409 |

Conjugacy Classes | p. 409 |

The Class Equation | p. 410 |

The Probability That Two Elements Commute | p. 411 |

The Sylow Theorems | p. 412 |

Applications of Sylow Theorems | p. 417 |

Exercises | p. 421 |

Biography of Ludwig Sylow | p. 427 |

Finite Simple Groups | p. 428 |

Historical Background | p. 428 |

Nonsimplicity Tests | p. 433 |

The Simplicity of A_{5} | p. 437 |

The Fields Medal | p. 438 |

The Cole Prize | p. 438 |

Exercises | p. 439 |

Biography of Michael Aschbacher | p. 442 |

Biography of Daniel Gorenstein | p. 443 |

Biography of John Thompson | p. 444 |

Generators and Relations | p. 445 |

Motivation | p. 445 |

Definitions and Notation | p. 446 |

Free Group | p. 447 |

Generators and Relations | p. 448 |

Classification of Groups of Order Up to 15 | p. 452 |

Characterization of Dihedral Groups | p. 454 |

Realizing the Dihedral Groups with Mirrors | p. 455 |

Exercises | p. 457 |

Biography of Marshall Hall, Jr. | p. 460 |

Symmetry Groups | p. 461 |

Isometries | p. 461 |

Classification of Finite Plane Symmetry Groups | p. 463 |

Classification of Finite Groups of Rotations in R^{3} | p. 464 |

Exercises | p. 466 |

Frieze Groups and Grystallographic Groups | p. 469 |

The Frieze Groups | p. 469 |

The Crystallographic Groups | p. 475 |

Identification of Plane Periodic Patterns | p. 481 |

Exercises | p. 487 |

Biography of M. C. Escher | p. 492 |

Biography of George Pólya | p. 493 |

Biography of John H. Conway | p. 494 |

Symmetry and Counting | p. 495 |

Motivation | p. 495 |

Burnside's Theorem | p. 496 |

Applications | p. 498 |

Group Action | p. 501 |

Exercises | p. 502 |

Biography of William Burnside | p. 505 |

Cayley Digraphs of Groups | p. 506 |

Motivation | p. 506 |

The Cayley Digraph of a Group | p. 506 |

Hamiltonian Circuits and Paths | p. 510 |

Some Applications | p. 516 |

Exercises | p. 519 |

Biography of William Rowan Hamilton | p. 524 |

Biography of Paul Erdos | p. 525 |

Introduction to Algebraic Coding Theory | p. 526 |

Motivation | p. 526 |

Linear Codes | p. 531 |

Parity-Check Matrix Decoding | p. 536 |

Coset Decoding | p. 539 |

Historical Note: The Ubiquitous Reed-Solomon Codes | p. 543 |

Exercises | p. 545 |

Biography of Richard W. Hamming | p. 550 |

Biography of Jessie MacWilliams | p. 551 |

Biography of Vera Pless | p. 552 |

An Introduction to Galois Theory | p. 553 |

Fundamental Theorem of Galois Theory | p. 553 |

Solvability of Polynomials by Radicals | p. 560 |

Insolvability of a Quintic | p. 564 |

Exercises | p. 565 |

Biography of Philip Hall | p. 569 |

Cyclotomic Extensions | p. 570 |

Motivation | p. 570 |

Cyclotomic Polynomials | p. 571 |

The Constructible Regular n-gons | p. 575 |

Exercises | p. 577 |

Biography of Carl Friedrich Gauss | p. 579 |

Biography of Manjul Bhargava | p. 580 |

Supplementary Exercises for Chapters 24-33 | p. 581 |

Selected Answers | p. A1 |

Index of Mathematicians | p. A45 |

Index of Terms | p. A47 |

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