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# Contemporary Abstract Algebra

**by**Gallian, Joseph

7th

### 9780547165097

0547165099

Hardcover

1/8/2009

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

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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