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Contemporary Abstract Algebra

by Unknown
Edition:
5th
ISBN13:

9780618122141

ISBN10:
0618122141
Format:
Paperback
Pub. Date:
7/1/2001
Publisher(s):
Brooks Cole
List Price: $97.66
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Summary

Joseph Gallian is a well-known active researcher and award-winning teacher. His Contemporary 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.

Table of Contents

Preface xi
PART 1 Integers and Equivalence Relations 1(28)
Preliminaries
3(26)
Properties of Integers
3(5)
Modular Arithmetic
8(6)
Mathematical Induction
14(3)
Equivalence Relations
17(3)
Functions (Mappings)
20(9)
Exercises
23(3)
Computer Exercises
26(3)
PART 2 Groups 29(198)
Introduction to Groups
31(11)
Symmetries of a Square
31(3)
The Dihedral Groups
34(8)
Exercises
37(4)
Biography of Niels Abel
41(1)
Groups
42(16)
Definition and Examples of Groups
42(8)
Elementary Properties of Groups
50(2)
Historical Note
52(6)
Exercises
53(4)
Computer Exercises
57(1)
Finite Groups; Subgroups
58(15)
Terminology and Notation
58(2)
Subgroup Tests
60(2)
Examples of Subgroups
62(11)
Exercises
67(4)
Computer Exercises
71(2)
Cyclic Groups
73(20)
Properties of Cyclic Groups
73(5)
Classification of Subgroups of Cyclic Groups
78(15)
Exercises
82(4)
Computer Exercises
86(1)
Biography of J. J. Sylvester
87(2)
Supplementary Exercises for Chapters 1--4
89(4)
Permutation Groups
93(25)
Definition and Notation
93(3)
Cycle Notation
96(2)
Properties of Permutations
98(10)
A Check-Digit Scheme Based on D5
108(10)
Exercises
111(4)
Computer Exercises
115(2)
Biography of Augustin Cauchy
117(1)
Isomorphisms
118(16)
Motivation
118(1)
Definition and Examples
119(3)
Cayley's Theorem
122(2)
Properties of Isomorphisms
124(2)
Automorphisms
126(8)
Exercises
129(4)
Biography of Arthur Cayley
133(1)
Cosets and Lagrange's Theorem
134(16)
Properties of Cosets
134(3)
Lagrange's Theorem and Consequences
137(3)
An Application of Cosets of Permutation Groups
140(2)
The Rotation Group of a Cube and a Soccer Ball
142(8)
Exercises
145(4)
Biography of Joseph Lagrange
149(1)
External Direct Products
150(22)
Definition and Examples
150(1)
Properties of External Direct Products
151(3)
The Group of Units Modulo n as an External Direct Product
154(2)
Applications
156(16)
Exercises
162(3)
Computer Exercises
165(2)
Biography of Leonard Adleman
167(2)
Supplementary Exercises for Chapters 5--8
169(3)
Normal Subgroups and Factor Groups
172(22)
Normal Subgroups
172(2)
Factor Groups
174(5)
Applications of Factors Groups
179(3)
Internal Direct Products
182(12)
Exercises
186(6)
Biography of Evariste Galois
192(2)
Group Homomorphisms
194(17)
Definition and Examples
194(2)
Properties of Homomorphisms
196(4)
The First Isomorphism Theorem
200(11)
Exercises
205(5)
Biography of Camille Jordan
210(1)
Fundamental Theorem of Finite Abelian Groups
211(16)
The Fundamental Theorem
211(1)
The Isomorphism Classes of Abelian Groups
212(4)
Proof of the Fundamental Theorem
216(11)
Exercises
219(3)
Computer Exercises
222(2)
Supplementary Exercises for Chapters 9--11
224(3)
PART 3 Rings 227(106)
Introduction to Rings
229(11)
Motivation and Definition
229(1)
Examples of Rings
230(1)
Properties of Rings
231(1)
Subrings
232(8)
Exercises
234(3)
Computer Exercises
237(2)
Biography of I. N. Herstein
239(1)
Integral Domains
240(13)
Definition and Examples
240(2)
Fields
242(2)
Characteristic of a Ring
244(9)
Exercises
246(4)
Computer Exercises
250(2)
Biography of Nathan Jacobson
252(1)
Ideals and Factor Rings
253(17)
Ideals
253(1)
Factor Rings
254(4)
Prime Ideals and Maximal Ideals
258(12)
Exercises
260(4)
Biography of Richard Dedekind
264(1)
Biography of Emmy Noether
265(2)
Supplementary Exercises for Chapters 12--14
267(3)
Ring Homomorphisms
270(13)
Definition and Examples
270(3)
Properties of Ring Homomorphisms
273(3)
The Field of Quotients
276(7)
Exercises
277(6)
Polynomial Rings
283(12)
Notation and Terminology
283(3)
The Division Algorithm and Consequences
286(9)
Exercises
290(4)
Biography of Saunders Mac Lane
294(1)
Factorization of Polynomials
295(17)
Reducibility Tests
295(3)
Irreducibility Tests
298(5)
Unique Factorization in Z[x]
303(2)
Weird Dice: An Application of Unique Factorization
305(7)
Exercises
307(3)
Computer Exercises
310(2)
Divisibility in Integral Domains
312(21)
Irreducibles, Primes
312(3)
Historical Discussion of Fermat's Last Theorem
315(3)
Unique Factorization Domains
318(3)
Euclidean Domains
321(12)
Exercises
325(4)
Biography of Sophie Germain
329(1)
Biography of Andrew Wiles
330(1)
Supplementary Exercises for Chapters 15--18
331(2)
PART 4 Fields 333(60)
Vector Spaces
335(9)
Definition and Examples
335(1)
Subspaces
336(1)
Linear Independence
337(7)
Exercises
339(3)
Biography of Emil Artin
342(1)
Biography of Olga Taussky-Todd
343(1)
Extension Fields
344(17)
The Fundamental Theorem of Field Theory
344(2)
Splitting Fields
346(7)
Zeros of an Irreducible Polynomial
353(8)
Exercises
357(3)
Biography of Leopold Kronecker
360(1)
Algebraic Extensions
361(13)
Characterization of Extensions
361(2)
Finite Extensions
363(5)
Properties of Algebraic Extensions
368(6)
Exercises
369(3)
Biography of Irving Kaplansky
372(2)
Finite Fields
374(10)
Classification of Finite Fields
374(1)
Structure of Finite Fields
375(4)
Subfields of a Finite Field
379(5)
Exercises
381(2)
Biography of L. E. Dickson
383(1)
Geometric Constructions
384(9)
Historical Discussion of Geometric Constructions
384(1)
Constructible Numbers
385(2)
Angle-Trisectors and Circle-Squarers
387(6)
Exercises
387(3)
Supplementary Exercises for Chapters 19--23
390(3)
PART 5 Special Topics 393(2)
Sylow Theorems
395(18)
Conjugacy Classes
395(1)
The Class Equation
396(1)
The Probability That Two Elements Commute
397(1)
The Sylow Theorems
398(5)
Applications of Sylow Theorems
403(10)
Exercises
407(5)
Biography of Ludvig Sylow
412(1)
Finite Simple Groups
413(20)
Historical Background
413(4)
Nonsimplicity Tests
417(6)
The Simplicity of A5
423(1)
The Fields Medal
424(1)
The Cole Prize
424(9)
Exercises
424(2)
Computer Exercises
426(2)
Biography of Michael Aschbacher
428(1)
Biography of Daniel Gorenstein
429(2)
Biography of John Thompson
431(2)
Generators and Relations
433(17)
Motivation
433(1)
Definitions and Notation
434(1)
Free Group
435(1)
Generators and Relations
436(4)
Classification of Groups of Order Up to 15
440(2)
Characterization of Dihedral Groups
442(2)
Realizing the Dihedral Groups with Mirrors
444(6)
Exercises
445(3)
Biography of Marshall Hall, Jr.
448(2)
Symmetry Groups
450(8)
Isometries
450(2)
Classification of Finite Plane Symmetry Groups
452(2)
Classification of Finite Groups of Rotations in R3
454(4)
Exercises
455(3)
Frieze Groups and Crystallographic Groups
458(27)
The Frieze Groups
458(5)
The Crystallographic Groups
463(2)
Identification of Plane Periodic Patterns
465(20)
Exercises
476(5)
Biography of M. C. Escher
481(1)
Biography of George Polya
482(1)
Biography of John H. Conway
483(2)
Symmetry and Counting
485(12)
Motivation
485(2)
Burnside's Theorem
487(1)
Applications
488(4)
Group Action
492(5)
Exercises
493(2)
Biography of William Burnside
495(2)
Cayley Digraphs of Groups
497(23)
Motivation
497(1)
The Cayley Digraph of a Group
497(4)
Hamiltonian Circuits and Paths
501(7)
Some Applications
508(12)
Exercises
511(5)
Biography of William Rowan Hamilton
516(2)
Biography of Paul Erdos
518(2)
Introduction to Algebraic Coding Theory
520(27)
Motivation
520(5)
Linear Codes
525(5)
Parity-Check Matrix Decoding
530(3)
Coset Decoding
533(4)
Historical Note: Reed-Solomon Codes
537(10)
Exercises
539(5)
Biography of Richard W. Hamming
544(1)
Biography of Jessie MacWilliams
545(1)
Biography of Vera Pless
546(1)
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(5)
Exercises
560(3)
Biography of Philip Hall
563(1)
Cyclotomic Extensions
564(1)
Motivation
564(1)
Cyclotomic Polynomials
565(4)
The Constructible Regular n-gons
569(2)
Exercises
571(1)
Computer Exercise
572(1)
Biography of Carl Friedrich Gauss
573(2)
Supplementary Exercises for Chapters 24--33
575
Selected Answers 1(40)
Text Credits 41(2)
Photo Credits 43(1)
Index of Mathematicians 44(2)
Index of Terms 46


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