A Course in Algebra

by Fan, Yun; Xiong, Q. Y.; Zheng, Y. L.

• ISBN13: 9789810240615
• ISBN10: 9810240619
• Format: Hardcover
• Copyright: 1999-12-01
• Publisher: WORLD SCIENTIFIC PUB CO INC

Summary

This volume is based on the lectures given by the authors at Wuhan University and Hubei University in courses on abstract algebra. It presents the fundamental concepts and basic properties of groups, rings, modules and fields, including the interplay between them and other mathematical branches and applied aspects.

Table of Contents

Preface	v
Preliminaries	1	(18)
Sets	1	(3)
Logic	4	(3)
Relations	7	(4)
Maps	11	(7)
Zorn's Lemma	18	(1)
Groups	19	(52)
Transformations and Permutations	19	(8)
Groups	27	(8)
Subgroups	35	(4)
Homomorphisms, Isomorphisms	39	(6)
Cosets	45	(4)
Normal Subgroups, Quotient Groups	49	(3)
Homomorphism Theorems	52	(4)
Cyclic Groups, Orders of Elements	56	(6)
Direct Products	62	(9)
Rings	71	(52)
Fundamentals	71	(6)
Zero Divisors, Invertible Elements	77	(4)
Ideals, Residue Rings	81	(5)
Homomorphism Theorems	86	(3)
Prime Ideals, Maximal Ideals	89	(4)
Direct Sums	93	(4)
Fraction Fields of Integral Domains	97	(4)
Polynomial Rings	101	(9)
Factorial Domains	110	(7)
Polynomial Rings over Factorial Rings	117	(6)
Modules	123	(60)
Modules	123	(4)
Homomorphisms	127	(6)
Direct Products, Direct Sums	133	(4)
Exact Sequences of Homomorphisms	137	(5)
Free Modules, Matrices over Rings	142	(8)
Matrices over Division Rings	150	(9)
Matrices over Commutative Rings	159	(5)
Algebras over Commutative Rings	164	(7)
Tensor Products	171	(5)
Projective Modules, Injective Modules	176	(7)
Fields	183	(58)
Subfields and Extensions	183	(4)
Single Extensions	187	(5)
Algebraic Extensions	192	(4)
Splitting Fields, Normal Extensions	196	(6)
Two Applications	202	(5)
Separability, Multiple Roots	207	(8)
Finite Fields	215	(5)
Coding	220	(9)
p-adic Numbers	229	(9)
Quaternions	238	(3)
Index	241