Contemporary Abstract Algebra

by Gallian, Joseph A.

• ISBN13: 9780395861790
• ISBN10: 0395861799
• Edition: 4th
• Format: Paperback
• Copyright: 1997-09-29
• Publisher: Brooks Cole

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	(26)
CHAPTER 0 Preliminaries	3	(24)
Properties of Integers	3	(5)
Modular Arithmetic	8	(5)
Mathematical Induction	13	(3)
Equivalence Relations	16	(2)
Functions (Mappings)	18	(3)
Exercises	21	(3)
Computer Exercises	24	(3)
PART 2 Groups	27	(196)
CHAPTER 1 Introduction to Groups	29	(11)
Symmetries of a Square	29	(3)
The Dihedral Groups	32	(3)
Exercises	35	(4)
Biography of Niels Abel	39	(1)
CHAPTER 2 Groups	40	(16)
Definition and Examples of Groups	40	(8)
Elementary Properties of Groups	48	(2)
Historical Note	50	(1)
Exercises	51	(4)
Computer Exercises	55	(2)
CHAPTER 3 Finite Groups; Subgroups	56	(15)
Terminology and Notation	56	(2)
Subgroup Tests	58	(2)
Examples of Subgroups	60	(5)
Exercises	65	(4)
Computer Exercises	69	(2)
CHAPTER 4 Cyclic Groups	71	(15)
Properties of Cyclic Groups	71	(4)
Classification of Subgroups of Cyclic Groups	75	(4)
Exercises	79	(3)
Computer Excercises	82	(2)
Biography of J.J. Sylvester	84	(2)
Supplementary Exercises for Chapters 1-4	86	(4)
CHAPTER 5 Permutation Groups	90	(25)
Definition and Notation	90	(3)
Cycle Notation	93	(2)
Properties of Permutations	95	(9)
A Check-Digit Scheme Based on D(5)	104	(3)
Exercises	107	(3)
Computer Exercises	110	(3)
Biography of Augustin Cauchy	113	(2)
CHAPTER 6 Isomorphisms	115	(17)
Motivation	115	(1)
Definition and Examples	116	(3)
Cayley's Theorem	119	(2)
Properties of Isomorphisms	121	(1)
Automorphisms	122	(4)
Exercises	126	(4)
Biography of Arthur Cayley	130	(2)
CHAPTER 7 Cosets and Lagrange's Theorem	132	(17)
Properties of Cosets	132	(3)
Lagrange's Theorem and Consequences	135	(2)
An Application of Cosets to Permutation Groups	137	(2)
The Rotation Group of a Cube and a Soccer Ball	139	(3)
Exercises	142	(5)
Biography of Joseph Lagrange	147	(2)
CHAPTER 8 External Direct Products	149	(19)
Definition and Examples	149	(2)
Properties of External Direct Products	151	(3)
The Group of Units Modulo n as an External Direct Product	154	(2)
Applications	156	(5)
Exercises	161	(3)
Computer Exercises	164	(2)
Biography of Leonard Adleman	166	(2)
Supplementary Exercises for Chapters 5-8	168	(3)
CHAPTER 9 Normal Subgroups and Factor Groups	171	(21)
Normal Subgroups	171	(1)
Factor Groups	172	(6)
Applications of Factor Groups	178	(3)
Internal Direct Products	181	(4)
Exercises	185	(5)
Biography of Evariste Galois	190	(2)
CHAPTER 10 Group Homomorphisms	192	(17)
Definition and Examples	192	(2)
Properties of Homomorphisms	194	(4)
The First Isomorphism Theorem	198	(5)
Exercises	203	(4)
Biography of Camille Jordan	207	(2)
CHAPTER 11 Fundamental Theorem of Finite Abelian Groups	209	(12)
The Fundamental Theorem	209	(1)
The Isomorphism Classes of Abelian Groups	210	(4)
Proof of the Fundamental Theorem	214	(3)
Exercises	217	(2)
Computer Exercises	219	(2)
Supplementary Exercises for Chapters 9-11	221	(2)
PART 3 Rings	223	(106)
CHAPTER 12 Introduction to Rings	225	(11)
Motivation and Definition	225	(1)
Examples of Rings	226	(1)
Properties of Rings	227	(1)
Subrings	228	(5)
Exercises	233	(1)
Computer Exercises	233	(2)
Biography of I.N. Herstein	235	(1)
CHAPTER 13 Integral Domains	236	(12)
Definition and Examples	236	(2)
Fields	238	(2)
Characteristic of a Ring	240	(2)
Exercises	242	(4)
Computer Exercises	246	(1)
Biography of Nathan Jacobson	247	(1)
CHAPTER 14 Ideals and Factor Rings	248	(15)
Ideals	248	(1)
Factor Rings	249	(4)
Prime Ideals and Maximal Ideals	253	(2)
Exercises	255	(4)
Biography of Richard Dedekind	259	(2)
Biography of Emmy Noether	261	(2)
Supplementary Exercises for Chapters 12-14	263	(3)
CHAPTER 15 Ring Homomorphisms	266	(13)
Definition and Examples	266	(3)
Properties of Ring Homomorphisms	269	(3)
The Field of Quotients	272	(2)
Exercises	274	(5)
CHAPTER 16 Polynomial Rings	279	(12)
Notation and Terminology	279	(3)
The Division Algorithm and Consequences	282	(4)
Exercises	286	(4)
Biography of Saunders Mac Lane	290	(1)
CHAPTER 17 Factorization of Polynomials	291	(17)
Reducibility Tests	291	(3)
Irreducibility Tests	294	(5)
Unique Factorization in Z[x]	299	(2)
Weird Dice: An Application of Unique Factorization	301	(2)
Exercises	303	(3)
Computer Exercises	306	(2)
CHAPTER 18 Divisibility in Integral Domains	308	(19)
Irreducibles, Primes	308	(3)
Historical Discussion of Fermat's Last Theorem	311	(3)
Unique Factorization Domains	314	(3)
Euclidean Domains	317	(3)
Exercises	320	(4)
Biography of Sophie Germain	324	(2)
Biography of Andrew Wiles	326	(1)
Supplementary Exercises for Chapters 15-18	327	(2)
PART 4 Fields	329	(64)
CHAPTER 19 Vector Spaces	331	(10)
Definition and Examples	331	(1)
Subspaces	332	(1)
Linear Independence	333	(2)
Exercises	335	(3)
Biography of Emil Artin	338	(2)
Biography of Olga Taussky-Todd	340	(1)
CHAPTER 20 Extension Fields	341	(18)
The Fundamental Theorem of Field Theory	341	(2)
Splitting Fields	343	(7)
Zeros of an Irreducible Polynomial	350	(4)
Exercises	354	(3)
Biography of Leopold Kronecker	357	(2)
CHAPTER 21 Algebraic Extensions	359	(13)
Characterization of Extensions	359	(2)
Finite Extensions	361	(4)
Properties of Algebraic Extensions	365	(2)
Exercises	367	(3)
Biography of Irving Kaplansky	370	(2)
CHAPTER 22 Finite Fields	372	(11)
Classification of Finite Fields	372	(1)
Structure of Finite Fields	373	(4)
Subfields of a Finite Field	377	(2)
Exercises	379	(2)
Biography of L.E. Dickson	381	(2)
CHAPTER 23 Geometric Constructions	383	(7)
Historical Discussion of Geometric Constructions	383	(1)
Constructible Numbers	384	(2)
Angle-Trisectors and Circle-Squarers	386	(1)
Exercises	387	(3)
Supplementary Exercises for Chapter 19-23	390	(3)
PART 5 Special Topics	393
CHAPTER 24 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	(4)
Exercises	407	(5)
Biography of Ludvig Sylow	412	(1)
CHAPTER 25 Finite Simple Groups	413	(21)
Historical Background	413	(4)
Nonsimplicity Tests	417	(6)
The Simplicity of A(5)	423	(1)
The Fields Medal	424	(1)
The Cole Prize	424	(1)
Exercises	424	(2)
Computer Exercises	426	(2)
Biography of Michael Aschbacher	428	(2)
Biography of Daniel Gorenstein	430	(2)
Biography of John Thompson	432	(2)
CHAPTER 26 Generators and Relations	434	(19)
Motivation	434	(1)
Definitions and Notation	435	(1)
Free Group	436	(1)
Generators and Relations	437	(4)
Classification of Groups of Order up to 15	441	(2)
Characterization of Dihedral Groups	443	(2)
Realizing the Dihedral Groups with Mirrors	445	(2)
Exercises	447	(4)
Biography of Marshall Hall, Jr.	451	(2)
CHAPTER 27 Symmetry Groups	453	(9)
Isometries	453	(2)
Classification of Finite Plane Symmetry Groups	455	(2)
Classification of Finite Groups of Rotations in R(3)	457	(2)
Exercises	459	(3)
CHAPTER 28 Frieze Groups and Crystallographic Groups	462	(28)
The Frieze Groups	462	(5)
The Crystallographic Groups	467	(2)
Identification of Plane Periodic Patterns	469	(11)
Exercises	480	(5)
Biography of M.C. Escher	485	(2)
Biography of George Polya	487	(1)
Biography of John H. Conway	488	(2)
CHAPTER 29 Symmetry and Counting	490	(12)
Motivation	490	(2)
Burnside's Theorem	492	(1)
Applications	493	(4)
Group Action	497	(1)
Exercises	498	(2)
Biography of William Burnside	500	(2)
CHAPTER 30 Cayley Digraphs of Groups	502	(23)
Motivation	502	(1)
The Cayley Digraphs of a Groups	502	(4)
Hamiltonian Circuits and Paths	506	(7)
Some Applications	513	(2)
Exercises	515	(6)
Biography of William Rowan Hamilton	521	(2)
Biography of Paul Erdos	523	(2)
CHAPTER 31 Introduction to Algebraic Coding Theory	525	(29)
Motivation	525	(5)
Linear Codes	530	(5)
Parity-Check Matrix Decoding	535	(3)
Coset Decoding	538	(4)
Historical Note: Reed-Solomon Codes	542	(2)
Exercises	544	(5)
Biography of Richard W. Hamming	549	(2)
Biography of Jessie MacWilliams	551	(1)
Biography of Vera Pless	552	(2)
CHAPTER 32 An Introduction to Galois Theory	554	(17)
Fundamental Theorem of Galois Theory	554	(7)
Solvability of Polynomials by Radicals	561	(5)
Insolvability of a Quintic	566	(1)
Exercises	567	(3)
Biography of Philip Hall	570	(1)
CHAPTER 33 Cyclotomic Extensions	571	(11)
Motivation	571	(1)
Cyclotomic Polynomials	572	(4)
The Constructible Regular n-gons	576	(2)
Exercises	578	(1)
Computer Exercise	579	(1)
Biography of Carl Friedrich Gauss	580	(2)
Supplementary Exercises for Chapters 24-33	582
Selected Answers	A1	(38)
Text Credits	A39	(2)
Photo Credits	A41	(2)
Notations	A43	(4)
Index of Mathematicians	A47	(2)
Index of Terms	A49