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Contemporary Abstract Algebra



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Brooks Cole
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CONTEMPORARY ABSTRACT ALGEBRA, EIGHTH EDITION provides a solid introduction to the traditional topics in abstract algebra while conveying to students that it is a contemporary subject used daily by working mathematicians, computer scientists, physicists, and chemists. The text includes numerous figures, tables, photographs, charts, biographies, computer exercises, and suggested readings giving the subject a current feel which makes the content interesting and relevant for students.

Table of Contents

Prefacep. xi
Integers and Equivalence Relationsp. 1
Preliminariesp. 3
Properties of Integersp. 3
Modular Arithmeticp. 6
Complex Numbersp. 13
Mathematical Inductionp. 14
Equivalence Relationsp. 17
Functions (Mappings)p. 20
Exercisesp. 23
Groupsp. 29
Introduction to Groupsp. 31
Symmetries of a Squarep. 31
The Dihedral Groupsp. 34
Exercisesp. 37
Biography of Niels Abelp. 41
Groupsp. 42
Definition and Examples of Groupsp. 42
Elementary Properties of Groupsp. 50
Historical Notep. 53
Exercisesp. 54
Finite Groups; Subgroupsp. 60
Terminology and Notationp. 60
Subgroup Testsp. 61
Examples of Subgroupsp. 65
Exercisesp. 68
Cyclic Groupsp. 77
Properties of Cyclic Groupsp. 77
Classification of Subgroups of Cyclic Groupsp. 82
Exercisesp. 87
Biography of James Joseph Sylvesterp. 93
Supplementary Exercises for Chapters 1-4p. 95
Permutation Groupsp. 99
Definition and Notationp. 99
Cycle Notationp. 102
Properties of Permutationsp. 104
A Check-Digit Scheme Based on D5p. 115
Exercisesp. 118
Biography of Augustin Cauchyp. 126
Isomorphismsp. 127
Motivationp. 127
Definition and Examplesp. 127
Cayley's Theoremp. 131
Properties of Isomorphismsp. 133
Automorphismsp. 134
Exercisesp. 138
Biography of Arthur Cayleyp. 143
Cosets and Lagrange's Theoremp. 144
Properties of Cosetsp. 144
Lagrange's Theorem and Consequencesp. 147
An Application of Cosets to Permutation Groupsp. 151
The Rotation Group of a Cube and a Soccer Ballp. 153
An Application of Cosets to the Rubik's Cubep. 155
Exercisesp. 156
Biography of Joseph Lagrangep. 161
External Direct Productsp. 162
Definition and Examplesp. 162
Properties of External Direct Productsp. 163
The Group of Units Modulo n as an External Direct Productp. 166
Applicationsp. 168
Exercisesp. 174
Biography of Leonard Adlemanp. 180
Supplementary Exercises for Chapters 5-8p. 181
Normal Subgroups and Factor Groupsp. 185
Normal Subgroupsp. 185
Factor Groupsp. 187
Applications of Factor Groupsp. 193
Internal Direct Productsp. 195
Exercisesp. 200
Biography of …variste Galoisp. 207
Group Homomorphismsp. 208
Definition and Examplesp. 208
Properties of Homomorphismsp. 210
The First Isomorphism Theoremp. 214
Exercisesp. 219
Biography of Camille Jordanp. 225
Fundamental Theorem of Finite Abelian Groupsp. 226
The Fundamental Theoremp. 226
The Isomorphism Classes of Abelian Groupsp. 226
Proof of the Fundamental Theoremp. 231
Exercisesp. 234
Supplementary Exercises for Chapters 9-11p. 238
Ringsp. 243
Introduction to Ringsp. 245
Motivation and Definitionp. 245
Examples of Ringsp. 246
Properties of Ringsp. 247
Subringsp. 248
Exercisesp. 250
Biography of I. N. Hersteinp. 254
Integral Domainsp. 255
Definition and Examplesp. 255
Fieldsp. 256
Characteristic of a Ringp. 258
Exercisesp. 261
Biography of Nathan Jacobsonp. 266
Ideals and Factor Ringsp. 267
Idealsp. 267
Factor Ringsp. 268
Prime Ideals and Maximal Idealsp. 272
Exercisesp. 274
Biography of Richard Dedekindp. 279
Biography of Emmy Noetherp. 280
Supplementary Exercises for Chapters 12-14p. 281
Ring Homomorphismsp. 285
Definition and Examplesp. 285
Properties of Ring Homomorphismsp. 288
The Field of Quotientsp. 290
Exercisesp. 292
Polynomial Ringsp. 298
Notation and Terminologyp. 298
The Division Algorithm and Consequencesp. 301
Exercisesp. 305
Biography of Saunders Mac Lanep. 310
Factorization of Polynomialsp. 311
Reducibility Testsp. 311
Irreducibility Testsp. 314
Unique Factorization in Z[x]p. 319
Weird Dice: An Application of Unique Factorizationp. 320
Exercisesp. 322
Biography of Serge Langp. 327
Divisibility in Integral Domainsp. 328
Irreducibles, Primesp. 328
Historical Discussion of Fermat's Last Theoremp. 331
Unique Factorization Domainsp. 334
Euclidean Domainsp. 337
Exercisesp. 341
Biography of Sophie Germainp. 345
Biography of Andrew Wilesp. 346
Supplementary Exercises for Chapters 15-18p. 347
Fieldsp. 349
Vector Spacesp. 351
Definition and Examplesp. 351
Subspacesp. 352
Linear Independencep. 353
Exercisesp. 355
Biography of Emil Artinp. 358
Biography of Olga Taussky-Toddp. 359
Extension Fieldsp. 360
The Fundamental Theorem of Field Theoryp. 360
Splitting Fieldsp. 362
Zeros of an Irreducible Polynomialp. 368
Exercisesp. 372
Biography of Leopold Kroneckerp. 375
Algebraic Extensionsp. 376
Characterization of Extensionsp. 376
Finite Extensionsp. 378
Properties of Algebraic Extensionsp. 382
Exercisesp. 384
Biography of Irving Kaplanskyp. 387
Finite Fieldsp. 388
Classification of Finite Fieldsp. 388
Structure of Finite Fieldsp. 389
Subfields of a Finite Fieldp. 393
Exercisesp. 395
Biography of L. E. Dicksonp. 398
Geometric Constructionsp. 399
Historical Discussion of Geometric Constructionsp. 399
Constructible Numbersp. 400
Angle-Trisectors and Circle-Squarersp. 402
Exercisesp. 402
Supplementary Exercises for Chapters 19-23p. 405
Special Topicsp. 407
Sylow Theoremsp. 409
Conjugacy Classesp. 409
The Class Equationp. 410
The Probability That Two Elements Commutep. 411
The Sylow Theoremsp. 412
Applications of Sylow Theoremsp. 417
Exercisesp. 421
Biography of Ludwig Sylowp. 427
Finite Simple Groupsp. 428
Historical Backgroundp. 428
Nonsimplicity Testsp. 433
The Simplicity of A5p. 437
The Fields Medalp. 438
The Cole Prizep. 438
Exercisesp. 439
Biography of Michael Aschbacherp. 442
Biography of Daniel Gorensteinp. 443
Biography of John Thompsonp. 444
Generators and Relationsp. 445
Motivationp. 445
Definitions and Notationp. 446
Free Groupp. 447
Generators and Relationsp. 448
Classification of Groups of Order Up to 15p. 452
Characterization of Dihedral Groupsp. 454
Realizing the Dihedral Groups with Mirrorsp. 455
Exercisesp. 457
Biography of Marshall Hall, Jr.p. 460
Symmetry Groupsp. 461
Isometriesp. 461
Classification of Finite Plane Symmetry Groupsp. 463
Classification of Finite Groups of Rotations in R3p. 464
Exercisesp. 466
Frieze Groups and Grystallographic Groupsp. 469
The Frieze Groupsp. 469
The Crystallographic Groupsp. 475
Identification of Plane Periodic Patternsp. 481
Exercisesp. 487
Biography of M. C. Escherp. 492
Biography of George Půlyap. 493
Biography of John H. Conwayp. 494
Symmetry and Countingp. 495
Motivationp. 495
Burnside's Theoremp. 496
Applicationsp. 498
Group Actionp. 501
Exercisesp. 502
Biography of William Burnsidep. 505
Cayley Digraphs of Groupsp. 506
Motivationp. 506
The Cayley Digraph of a Groupp. 506
Hamiltonian Circuits and Pathsp. 510
Some Applicationsp. 516
Exercisesp. 519
Biography of William Rowan Hamiltonp. 524
Biography of Paul Erdosp. 525
Introduction to Algebraic Coding Theoryp. 526
Motivationp. 526
Linear Codesp. 531
Parity-Check Matrix Decodingp. 536
Coset Decodingp. 539
Historical Note: The Ubiquitous Reed-Solomon Codesp. 543
Exercisesp. 545
Biography of Richard W. Hammingp. 550
Biography of Jessie MacWilliamsp. 551
Biography of Vera Plessp. 552
An Introduction to Galois Theoryp. 553
Fundamental Theorem of Galois Theoryp. 553
Solvability of Polynomials by Radicalsp. 560
Insolvability of a Quinticp. 564
Exercisesp. 565
Biography of Philip Hallp. 569
Cyclotomic Extensionsp. 570
Motivationp. 570
Cyclotomic Polynomialsp. 571
The Constructible Regular n-gonsp. 575
Exercisesp. 577
Biography of Carl Friedrich Gaussp. 579
Biography of Manjul Bhargavap. 580
Supplementary Exercises for Chapters 24-33p. 581
Selected Answersp. A1
Index of Mathematiciansp. A45
Index of Termsp. A47
Table of Contents provided by Ingram. All Rights Reserved.

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