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Contemporary Abstract Algebra

by
Edition:
7th
ISBN13:

9780547165097

ISBN10:
0547165099
Format:
Hardcover
Pub. Date:
1/8/2009
Publisher(s):
Cengage Learning
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Summary

The seventh edition of Contemporary Abstract Algebra, by Joseph A. Gallian, Provides a solid introduction to the traditional topics in abstract algebra while conveying that it is a contemporary subject used daily by working mathematicians, computer scientist, and chemists. The text includes numerous theoretical and computational exercises, figures, and tables to teach you how to work out problems, as well as to write proofs. Additionally, the author provides biographies, poems, song Lyrics, historical notes, and much more to make reading the text an interesting, accessible and enjoyable experience. Contemporary Abstract Algebra will keep you engaged and gives you a great introduction to an important subject.

Table of Contents

Prefacep. xi
Integers and Equivalence Relationsp. 1
Preliminariesp. 3
Properties of Integersp. 3
Madular Arithmeticp. 7
Mathematical Inductionp. 12
Equivalence Relationsp. 15
Functions (Mappings)p. 18
Exercisesp. 21
Computer Exercisesp. 25
Groupsp. 27
Introduction to Groupsp. 29
Symmetries of a Squarep. 29
The Dihedral Groupsp. 32
Exercisesp. 35
Biography of Niels Abelp. 39
Groupsp. 40
Definition and Examples of Groups o40
Elementary Properties of Groupsp. 48
Historical Notep. 51
Exercisesp. 52
Computer Exercisesp. 55
Finite Groups; Subgroupsp. 57
Terminology and Notationp. 57
Subgroup Testsp. 58
Examples of Subgroupsp. 61
Exercisesp. 64
Computer Exercisesp. 70
Cyclic Groupsp. 72
Properties of Cycle Groupsp. 72
Classification of Subgroups of Cyclic Groupsp. 77
Exercisesp. 81
Computer Exercisesp. 86
Biography of J. J. Sylvesterp. 89
Supplementary Exercises for Chapters1-4p. 91
Permutation Groupsp. 95
Difinition and Notationp. 95
Cycle Nationp. 98
Properties of Permutationsp. 100
A Check Digit Scheme Based on D5p. 110
Exercisesp. 113
Computer Exercisesp. 118
Biography of Augustin Cauchyp. 121
Isomorphismsp. 122
Motivationp. 122
Dfinition and Examplesp. 122
Cayley∆s Theoremp. 126
Properties of Isomorphismsp. 128
Automorphismsp. 129
Exercisesp. 133
Computer Exercisep. 136
Biography of Arthur Cayleyp. 137
Cosets and Lagrange∆s Theoremp. 138
Properties of Cosetsp. 138
Lagrange∆s Theorem and Consequencesp. 141
An Application of Cosets of Permutation Groupsp. 145
The Rotation Group of a Cube and a Soccer Ballp. 146
Exercisesp. 149
Computer Exercisep. 153
Biography of Joseph Lagrangep. 154
External Direct Productsp. 155
Definition and Examplesp. 155
Properties of External Direct Productsp. 156
The Group of Units Modulo n as an External Direct Productsp. 159
Applicationsp. 161
Exercisesp. 167
Computer Exercisesp. 170
Biorgaphy of Leonard Adlemanp. 173
Supplementary Exercises for Chapters 5-8p. 174
Normal Subgroups and Factor Groupsp. 178
Normal Subgroupsp. 178
Factor Groupsp. 180
Applicatons of Factor Groupsp. 185
Internal Direct Productsp. 188
Exercisesp. 193
Biography of Evariste Galoisp. 199
Group Homomorphismsp. 200
Difinition and Examplesp. 200
Properties Of Homomorphismsp. 202
The First Isomorphism Theoremp. 206
Exercisesp. 211
Computer Exercisep. 216
Biography of Camille Jordanp. 217
Fundamental Theorem of Finite Abelian Groupsp. 218
The Fundamental Theoremp. 218
The Isomorphism Classes of Abelian Groupsp. 218
Proof of the Fundamental Theoremp. 223
Exercisesp. 226
Computer Exercisesp. 228
Supplementary Exercises for Chapter 9-11p. 230
Ringsp. 235
Introduction to Ringsp. 237
Motivation and Definitionp. 237
Examples of Ringsp. 238
Properties of Ringsp. 239
Subringsp. 240
Exercisesp. 242
Computer Exercisesp. 245
Biography of I. N. Hersteinp. 248
Integral Domainsp. 249
Definition and Examplesp. 249
Fieldsp. 250
Characteristic of a Ringp. 225
Exercisesp. 255
Computer Exercisesp. 259
Biography of Nathan Jacobsonp. 261
Ideals and Factor Ringsp. 262
Idealsp. 262
Factor Ringsp. 263
Prime Ideals and Maximal Idealsp. 267
Exercisesp. 269
Computer Exercisesp. 273
Biography of Richard Dedekindp. 274
Biography of Emmy Noetherp. 275
Supplementary Exercises for Chapters 12-14p. 276
Ring Homomorphismsp. 280
Definition and Examplep. 280
Properties of Ring Homomorphismsp. 283
The Field of Quotientsp. 285
Exercisesp. 287
Polynomial Ringsp. 293
Notation and Terminologyp. 293
The Division Algorithm and Consequencesp. 296
Exercisesp. 300
Biography of Sounders Mac Lanep. 304
Factorization of Polynomialsp. 305
Reducibility Testsp. 305
Irreducibility Testsp. 308
Unique Factorization in Z[x]p. 313
Weird Dice: An Application of Unique Factorizationp. 314
Exercisesp. 316
Computer Exercisesp. 319
Biography of Serge Langp. 321
Divisibility in Integral Domainsp. 322
Irreducibles, Primesp. 322
Historical Discussion of Fermat∆s Last Theoremp. 325
Unique Factorization Domainsp. 328
Euclidean Domainsp. 331
Exercisesp. 335
Comupter Exercisep. 337
Biography of Sophie Germainp. 339
Biography of Andrew Wilesp. 340
Supplementary Exercises for Chapters 15-18p. 341
Fieldsp. 343
Vector Spacesp. 345
Definition and Examplesp. 345
Subspaces
Linear Independencep. 347
Exercisesp. 349
Biography of Emil Artinp. 352
Biography of Olga Taussky-Toddp. 353
Extension Fieldsp. 354
The Fundamental Theorem of Field theoryp. 354
Splitting Fieldsp. 356
Zeros of an Irreducible Polynomialp. 362
Exercisesp. 366
Biography of Leopold Kroneckerp. 369
Algebraci Extensionsp. 370
Characterization of Extensionsp. 370
Finite Extensionsp. 372
Properties of Algebraic Extensionsp. 376
Exercisesp. 378
Biography of Irving Kaplanskyp. 381
Finite Fieldsp. 382
Classification of Finite Fieldsp. 382
Struction of Finite Fieldsp. 383
Subfields of a Finite Fieldp. 387
Exercisesp. 389
Computer Exercisesp. 391
Biography of L. E. Dicksonp. 392
Geometric Constructionsp. 393
Historical Discussion of Geometric Constructionsp. 393
Constructible Numbersp. 394
Angle-Trisectors and Circle-Squarersp. 396
Exercisesp. 396
Supplementary Exercises for Chaptersp. 19-23
Special Topicsp. 401
Sylow Theoremsp. 403
Conjugacy Classesp. 403
The Class Equationp. 404
The Probability That Two Elements Commutep. 405
The Sylow Theoremsp. 406
Applications of Sylow Theoremsp. 411
Exercisesp. 414
Computer Exercisep. 418
Biography of Ludwig Sylowp. 419
Finite Simple Groupsp. 420
Historical Backgroundp. 420
Nonsimplicity Testsp. 245
The Simplicity of A5p. 429
The Fields Medalp. 430
The Cole Prizep. 430
Execisesp. 431
Computer Exercisesp. 432
Biography of Michael Aschbacherp. 434
Biography of Daniel Gorensteinp. 435
Biography of John Thompsonp. 436
Generators and Relationsp. 437
Motivationp. 437
Definitions and Notationp. 438
Free Groupp. 439
Generators and Relationsp. 440
Classification of Groups of Order Up to 15p. 444
Characterization of Dihedral Groupp. 446
Realizing the Dihedral Groups with Mirrorsp. 447
Exercisesp. 449
Biography of Marshall Hall, Jr.p. 452
Symmetry Groupsp. 453
Isometriesp. 453
Classification of Finite Plane Symmetry Groupp. 455
Classification of Finite Groups of Rotations in R3p. 456
Exercisesp. 458
Frieze Groups and Crystallographic Groupsp. 461
The Frieze Groupsp. 461
The Crystallographic Groupsp. 467
Identification of Plane Periodic Patternsp. 473
Exercisesp. 479
Biography of M. C. Escherp. 484
Biography of George Polyap. 485
Biography of John H. Conwayp. 486
Symmetry and Countingp. 487
Motivationp. 487
Burnside∆s Theoremp. 488
Applicationsp. 490
Group Actionp. 493
Exercisesp. 494
Biography of William Burnsidep. 497
Cayley Digraphs of Groupsp. 498
Motivatonp. 498
The Cayley Digraph of a Groupp. 498
Hamiltonian Circuits and Pathsp. 502
Some Apllicationsp. 508
Exercisesp. 511
Biography of William Rowan Hamiltonp. 516
Biography of Paul Erdosp. 517
Indtoduction to Algebraic Coding Theoryp. 518
Motivationp. 518
Liner Codesp. 523
Parity-Check Matrix Decodingp. 528
Coset Decodingp. 531
Hestorical Note: The Ubiquitous Reed-Solomon Codesp. 535
Exercisesp. 537
Biography of Richard W. Hammingp. 542
Biography of Jessie Mac Williamsp. 543
Biography of Vera Plessp. 544
An Introduction to Galois Theoryp. 545
Fundamental Theorem of Galois Theoryp. 545
Solvability of Polynomials by Radicalsp. 552
Insolvability of a Quinticp. 556
Exercisesp. 557
Biography of Philip Hallp. 560
Cyclotomic Extensionsp. 561
Motivationp. 561
Cyclotomic Polynomialsp. 562
The Constructible Regular n-Gonsp. 566
Exercisesp. 568
Computer Exercisp. 569
Biography of Carl Friedrich Gaussp. 570
Biography of Manjul Bhargavap. 571
Supplementary Exercises for Chapters 24-33p. 572
Selected Answersp. A1
Text Creditsp. A40
Photo Creditsp. A42
Index of Mathematiciansp. A43
Index of Termsp. A45
Table of Contents provided by Ingram. All Rights Reserved.


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