Rent 9780547165097 | Contemporary Abstract Algebra

Summary Table of Contents

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.

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
... MOREIntroduction 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
Cayley�s 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 Lagrange�s Theorem	p. 138
Properties of Cosets	p. 138
Lagrange�s 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 Fermat�s 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
Burnside�s 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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