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Contemporary Abstract Algebra

by
Edition:
6th
ISBN13:

9780618514717

ISBN10:
0618514716
Format:
Hardcover
Pub. Date:
12/15/2004
Publisher(s):
Cengage Learning
List Price: $260.95
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Summary

Joseph Gallian is a well-known active researcher and award-winning teacher. HisContemporary Abstract Algebra,6/e, includes challenging topics in abstract algebra as well as numerous figures, tables, photographs, charts, biographies, computer exercises, and suggested readings that give the subject a current feel and makes the content interesting and relevant for students. Updated!This edition includes many new exercises and computer exercises. Updated!Biographies, quotations, and suggested readings have been updated for currency and relevance.

Table of Contents

Preface ix
PART 1 Integers and Equivalence Relations 1(28)
CHAPTER 0 Preliminaries
3(26)
Properties of Integers
3(5)
Modular Arithmetic
8(6)
Mathematical Induction
14(3)
Equivalence Relations
17(3)
Functions (Mappings)
20(3)
Exercises
23(3)
Computer Exercises
26(3)
PART 2 Groups 29(204)
CHAPTER 1 Introduction to Groups
31(11)
Symmetries of a Square
31(3)
The Dihedral Groups
34(3)
Exercises
37(5)
Biography of Niels Abel
41(1)
CHAPTER 2 Groups
42(17)
Definition and Examples of Groups
42(8)
Elementary Properties of Groups
50(2)
Historical Note
52(1)
Exercises
53(4)
Computer Exercises
57(2)
CHAPTER 3 Finite Groups; Subgroups
59(14)
Terminology and Notation
59(2)
Subgroup Tests
61(2)
Examples of Subgroups
63(4)
Exercises
67(4)
Computer Exercises
71(2)
CHAPTER 4 Cyclic Groups
73(17)
Properties of Cyclic Groups
73(5)
Classification of Subgroups of Cyclic Groups
78(4)
Exercises
82(4)
Computer Exercises
86(8)
Biography of J.J. Sylvester
88(2)
Supplementary Exercises for Chapters 1- 4
90(4)
CHAPTER 5 Permutation Groups
94(26)
Definition and Notation
94(3)
Cycle Notation
97(2)
Properties of Permutations
99(10)
A Check-Digit Scheme Based on D5
109(3)
Exercises
112(4)
Computer Exercises
116(4)
Biography of Augustin Cauchy
119(1)
CHAPTER 6 Isomorphisms
120(17)
Motivation
120(1)
Definition and Examples
121(3)
Cayley's Theorem
124(2)
Properties of Isomorphisms
126(2)
Automorphisms
128(4)
Exercises
132(5)
Biography of Arthur Cayley
136(1)
CHAPTER 7 Cosets and Lagrange's Theorem
137(16)
Properties of Cosets
137(3)
Lagrange's Theorem and Consequences
140(3)
An Application of Cosets to Permutation Groups
143(2)
The Rotation Group of a Cube and a Soccer Ball
145(3)
Exercises
148(3)
Computer Exercises
151(2)
Biography of Joseph Lagrange
152(1)
CHAPTER 8 External Direct Products
153(21)
Definition and Examples
153(1)
Properties of External Direct Products
154(3)
The Group of Units Modulo n as an External Direct Product
157(2)
Applications
159(6)
Exercises
165(3)
Computer Exercises
168(9)
Biography of Leonard Adleman
172(2)
Supplementary Exercises for Chapters 5-8
174(3)
CHAPTER 9 Normal Subgroups and Factor Groups
177(22)
Normal Subgroups
177(2)
Factor Groups
179(5)
Applications of Factor Groups
184(3)
Internal Direct Products
187(4)
Exercises
191(8)
Biography of Évariste Galois
197(2)
CHAPTER 10 Group Homomorphisms
199(18)
Definition and Examples
199(2)
Properties of Homomorphisms
201(4)
The First Isomorphism Theorem
205(5)
Exercises
210(4)
Computer Exercises
214(3)
Biography of Camille Jordan
216(1)
CHAPTER 11 Fundamental Theorem of Finite Abelian Groups
217(13)
The Fundamental Theorem
217(1)
The Isomorphism Classes of Abelian Groups
218(4)
Proof of the Fundamental Theorem
222(3)
Exercises
225(3)
Computer Exercises
228(2)
Supplementary Exercises for Chapters 9-11
230(3)
PART 3 Rings 233(108)
CHAPTER 12 Introduction to Rings
235(13)
Motivation and Definition
235(1)
Examples of Rings
236(1)
Properties of Rings
237(1)
Subrings
238(2)
Exercises
240(3)
Computer Exercises
243(5)
Biography of I.N. Herstein
247(1)
CHAPTER 13 Integral Domains
248(13)
Definition and Examples
248(2)
Fields
250(2)
Characteristic of a Ring
252(2)
Exercises
254(4)
Computer Exercises
258(3)
Biography of Nathan Jacobson
260(1)
CHAPTER 14 Ideals and Factor Rings
261(14)
Ideals
261(1)
Factor Rings
262(4)
Prime Ideals and Maximal Ideals
266(2)
Exercises
268(4)
Computer Exercises
272(6)
Biography of Richard Dedekind
273(1)
Biography of Emmy Noether
274(1)
Supplementary Exercises for Chapters 12-14
275(3)
CHAPTER 15 Ring Homomorphisms
278(13)
Definition and Examples
278(3)
Properties of Ring Homomorphisms
281(3)
The Field of Quotients
284(2)
Exercises
286(5)
CHAPTER 16 Polynomial Rings
291(12)
Notation and Terminology
291(3)
The Division Algorithm and Consequences
294(4)
Exercises
298(5)
Biography of Saunders Mac Lane
302(1)
CHAPTER 17 Factorization of Polynomials
303(17)
Reducibility Tests
303(3)
Irreducibility Tests
306(5)
Unique Factorization in Z[x]
311(2)
Weird Dice: An Application of Unique Factorization
313(2)
Exercises
315(3)
Computer Exercises
318(2)
CHAPTER 18 Divisibility in Integral Domains
320(19)
Irreducibles, Primes
320(3)
Historical Discussion of Fermat's Last Theorem
323(3)
Unique Factorization Domains
326(3)
Euclidean Domains
329(4)
Exercises
333(2)
Computer Exercises
335(8)
Biography of Sophie Germain
337(1)
Biography of Andrew Wiles
338(1)
Supplementary Exercises for Chapters 15-18
339(2)
PART 4 Fields 341(58)
CHAPTER 19 Vector Spaces
343(9)
Definition and Examples
343(1)
Subspaces
344(1)
Linear Independence
345(2)
Exercises
347(5)
Biography of Emil Artin
350(1)
Biography of Olga Taussky-Todd
351(1)
CHAPTER 20 Extension Fields
352(17)
The Fundamental Theorem of Field Theory
352(2)
Splitting Fields
354(7)
Zeros of an Irreducible Polynomial
361(4)
Exercises
365(4)
Biography of Leopold Kronecker
368(1)
CHAPTER 21 Algebraic Extensions
369(12)
Characterization of Extensions
369(2)
Finite Extensions
371(4)
Properties of Algebraic Extensions
375(2)
Exercises
377(4)
Biography of Irving Kaplansky
380(1)
CHAPTER 22 Finite Fields
381(10)
Classification of Finite Fields
381(1)
Structure of Finite Fields
382(4)
Subfields of a Finite Field
386(2)
Exercises
388(1)
Computer Exercises
389(2)
Biography of L.E. Dickson
390(1)
CHAPTER 23 Geometric Constructions
391(6)
Historical Discussion of Geometric Constructions
391(1)
Constructible Numbers
392(2)
Angle-Trisectors and Circle-Squarers
394(1)
Exercises
394(3)
Supplementary Exercises for Chapters 19-23
397(2)
PART 5 Special Topics 399
CHAPTER 24 Sylow Theorems
401(18)
Conjugacy Classes
401(1)
The Class Equation
402(1)
The Probability That Two Elements Commute
403(1)
The Sylow Theorems
404(6)
Applications of Sylow Theorems
410(3)
Exercises
413(6)
Biography of Ludwig Sylow
418(1)
CHAPTER 25 Finite Simple Groups
419(18)
Historical Background
419(5)
Nonsimplicity Tests
424(4)
The Simplicity of A5
428(1)
The Fields Medal
429(1)
The Cole Prize
430(1)
Exercises
430(1)
Computer Exercises
431(6)
Biography of Michael Aschbacher
434(1)
Biography of Daniel Gorenstein
435(1)
Biography of John Thompson
436(1)
CHAPTER 26 Generators and Relations
437(17)
Motivation
437(1)
Definitions and Notation
438(1)
Free Group
439(1)
Generators and Relations
440(4)
Classification of Groups of Order Up to 15
444(2)
Characterization of Dihedral Groups
446(2)
Realizing the Dihedral Groups with Mirrors
448(1)
Exercises
449(5)
Biography of Marshall Hall, Jr.
453(1)
CHAPTER 27 Symmetry Groups
454(8)
Isometries
454(2)
Classification of Finite Plane Symmetry Groups
456(2)
Classification of Finite Groups of Rotations in R3
458(1)
Exercises
459(3)
CHAPTER 28 Frieze Groups and Crystallographic Groups
462(26)
The Frieze Groups
462(6)
The Crystallographic Groups
468(2)
Identification of Plane Periodic Patterns
470(10)
Exercises
480(8)
Biography of M.C. Escher
485(1)
Biography of George Pólya
486(1)
Biography of John H. Conway
487(1)
CHAPTER 29 Symmetry and Counting
488(12)
Motivation
488(2)
Burnside's Theorem
490(1)
Applications
491(4)
Group Action
495(1)
Exercises
496(4)
Biography of William Burnside
498(2)
CHAPTER 30 Cayley Digraphs of Groups
500(21)
Motivation
500(1)
The Cayley Digraph of a Group
500(4)
Hamiltonian Circuits and Paths
504(7)
Some Applications
511(3)
Exercises
514(7)
Biography of William Rowan Hamilton
519(1)
Biography of Paul Erdos
520(1)
CHAPTER 31 Introduction to Algebraic Coding Theory
521(26)
Motivation
521(5)
Linear Codes
526(5)
Parity-Check Matrix Decoding
531(3)
Coset Decoding
534(4)
Historical Note: Reed-Solomon Codes
538(2)
Exercises
540(7)
Biography of Richard W. Hamming
544(1)
Biography of Jessie MacWilliams
545(1)
Biography of Vera Pless
546(1)
CHAPTER 32 An Introduction to Galois Theory
547(17)
Fundamental Theorem of Galois Theory
547(7)
Solvability of Polynomials by Radicals
554(5)
Insolvability of a Quintic
559(1)
Exercises
560(4)
Biography of Philip Hall
563(1)
CHAPTER 33 Cyclotomic Extensions
564(11)
Motivation
564(1)
Cyclotomic Polynomials
565(4)
The Constructible Regular n-gons
569(2)
Exercises
571(1)
Computer Exercise
572(3)
Biography of Carl Friedrich Gauss
573(1)
Biography of Manful Bhargava
574(1)
Supplementary Exercises for Chapters 24-33
575
Selected Answers A1
Text Credits A44
Photo Credits A46
Index of Mathematicians A47
Index of Terms A49


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