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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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