A First Course in Abstract Algebra

by Fraleigh, John B.

• ISBN13: 9780201335965
• ISBN10: 0201335964
• Additional ISBN(s): 9780201763904
• Edition: 6th
• Format: Hardcover
• Copyright: 1999-01-01
• Publisher: Addison Wesley
• More Book Details
• Purchase Benefits

Summary

Considered a classic by many, A First Course in Abstract Algebra is an in-depth, introductory text which gives students a firm foundation for more specialized work by emphasizing an understanding of the nature of algebraic structures. The Sixth Edition continues its tradition of teaching in a classical manner, while integrating field theory and new exercises.

Table of Contents

0 A FEW PRELIMINARIES	1	(30)
0.1 Mathematics and Proofs	1	(6)
0.2 Sets and Relations	7	(10)
0.3 Mathematical Induction	17	(4)
0.4 Complex and Matrix Algebra	21	(10)
1 GROUPS AND SUBGROUPS	31	(62)
1.1 Binary Operations	31	(12)
*Finite-State Machines (Automata)	41	(2)
1.2 Isomorphic Binary Structures	43	(8)
1.3 Groups	51	(14)
1.4 Subgroups	65	(10)
1.5 Cyclic Groups and Generators	75	(18)
Cayley Digraphs	87	(6)
2 MORE GROUPS AND COSETS	93	(68)
2.1 Groups of Permutations	93	(14)
Automata	105	(2)
2.2 Orbits, Cycles, and the Alternating Groups	107	(13)
Plane Isometries	117	(3)
2.3 Cosets and the Theorem of Lagrange	120	(8)
2.4 Direct Products and Finitely Generated Abelian Groups	128	(20)
Periodic Functions	139	(2)
Plane Isometries	141	(7)
2.5 Binary Linear Codes	148	(13)
3 HOMOMORPHISMS AND FACTOR GROUPS	161	(48)
3.1 Homomorphisms	161	(11)
3.2 Factor Groups	172	(7)
3.3 Factor-Group Computations and Simple Groups	179	(11)
3.4 Series of Groups	190	(7)
3.5 Group Action on a Set	197	(7)
3.6 Applications of G-Sets to Counting	204	(5)
4 ADVANCED GROUP THEORY	209	(44)
4.1 Isomorphism Theorems: Proof of the Jordan-Holder Theorem	209	(8)
4.2 Sylow Theorems	217	(7)
4.3 Applications of the Sylow Theory	224	(6)
4.4 Free Abelian Groups	230	(8)
4.5 Free Groups	238	(6)
4.6 Group Presentations	244	(9)
5 INTRODUCTION TO RINGS AND FIELDS	253	(72)
5.1 Rings and Fields	253	(11)
5.2 Integral Domains	264	(7)
5.3 Fermat's and Euler's Theorems	271	(6)
5.4 The Field of Quotients of an Integral Domain	277	(8)
5.5 Rings of Polynomials	285	(12)
5.6 Factorization of Polynomials over a Field	297	(11)
5.7 Noncommutative Examples	308	(8)
5.8 Ordered Rings and Fields	316	(9)
6 FACTOR RINGS AND IDEALS	325	(30)
6.1 Homomorphisms and Factor Rings	325	(9)
6.2 Prime and Maximal Ideals	334	(10)
6.3 Grobner Bases for Ideals	344	(11)
7 FACTORIZATION	355	(28)
7.1 Unique Factorization Domains	355	(13)
7.2 Euclidian Domains	368	(7)
7.3 Gaussian Integers and Norms	375	(8)
8 EXTENSION FIELDS	383	(48)
8.1 Introduction to Extension Fields	383	(10)
8.2 Vector Spaces	393	(9)
8.3 Algebraic Extensions	402	(10)
8.4 Geometric Constructions	412	(7)
8.5 Finite Fields	419	(5)
8.6 Additional Algebraic Structures	424	(7)
9 AUTOMORPHISMS AND GALOIS THEORY	431	(64)
9.1 Automorphisms of Fields	431	(10)
9.2 The Isomorphism Extension Theorem	441	(7)
9.3 Splitting Fields	448	(5)
9.4 Separable Extensions	453	(8)
9.5 Totally Inseparable Extensions	461	(4)
9.6 Galois Theory	465	(9)
9.7 Illustrations of Galois Theory	474	(7)
9.8 Cyclotomic Extensions	481	(7)
9.9 Insolvability of the Quintic	488	(7)
BIBLIOGRAPHY	495	(4)
NOTATIONS	499	(4)
ANSWERS TO ODD-NUMBERED EXERCISES NOT ASKING FOR DEFINITIONS OR PROOFS	503	(22)
INDEX	525