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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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