Contemporary Abstract Algebra

by Gallian, Joseph

• ISBN13: 9781133599708
• ISBN10: 1133599702
• Additional ISBN(s): 9780367651787, 9780547165097
• Edition: 8th
• Format: Hardcover
• Copyright: 2012-07-09
• Publisher: Cengage Learning
• More Book Details
• Purchase Benefits

Summary

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

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 D5	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 A5	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 R3	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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