Introduction to Abstract Algebra, 2nd Edition

by W. Keith Nicholson

• ISBN13: 9780471331094
• ISBN10: 0471331090
• Edition: 2nd
• Format: Hardcover
• Copyright: 1999-07-01
• Publisher: Wiley-Interscience

Summary

The Third Edition of Introduction to Abstract Algebra continues to provide an accessible introduction to the basic structures of abstract algebra: groups, rings, and fields. The text's unique approach helps you advance to abstract theory by presenting concrete examples of induction, number theory, integers modulo n, and permutations before the abstract structures are defined.

Author Biography

W. KEITH NICHOLSON, PhD, is Professor of Mathematics at the University of Calgary, Calgary, Alberta, Canada.

Table of Contents

Preface	ix
Notation Used in the Text	xvi
A Sketch of the History of Algebra to 1929	xix
Preliminaries	1	(25)
Proofs	1	(5)
Sets	6	(5)
Mappings	11	(9)
Equivalences	20	(6)
Integers and Permutations	26	(53)
Induction	27	(9)
Divisibility and Prime Factorization	36	(12)
Integers Modulo n	48	(13)
Biography of Pierre de Fermat	57	(4)
Permutations	61	(15)
An Application to Cryptography1	76	(3)
Groups	79	(109)
Binary Operations	80	(8)
Groups	88	(11)
Subgroups	99	(6)
Cyclic Groups and the Order of an Element	105	(9)
Homomorphisms and Isomorphisms	114	(13)
Biography of Arthur Cayley	124	(3)
Cosets and Lagrange's Theorem	127	(12)
Biography of Joseph Louis Lagrange	135	(4)
Groups of Motions and Symmetries	139	(5)
Normal Subgroups	144	(10)
Factor Groups	154	(7)
The Isomorphism Theorem	161	(8)
An Application to Binary Linear Codes	169	(19)
Rings	188	(55)
Examples and Basic Properties	189	(14)
Integral Domains and Fields	203	(11)
Ideals and Factor Rings	214	(11)
Biography of Richard Dedekind	220	(5)
Homomorphisms	225	(15)
Biography of Emmy Noether	235	(5)
Ordered Integral Domains	240	(3)
Polynomials	243	(63)
Polynomials	244	(16)
Factorization of Polynomials over a Field	260	(17)
Biography of Carl Friedrich Gauss	272	(5)
Factor Rings of Polynomials over a Field	277	(11)
Partial Fractions	288	(3)
Symmetric Polynomials	291	(12)
Formal Construction of Polynomials	303	(3)
Factorization in Integral Domains	306	(27)
Irreducibles and Unique Factorization	307	(14)
Principal Ideal Domains	321	(12)
Biography of Ernst Eduard Kummer	329	(4)
Fields	333	(60)
Vector Spaces	334	(10)
Algebraic Extensions	344	(10)
Splitting Fields	354	(9)
Biography of Leopold Kronecker	361	(2)
Finite Fields	363	(7)
Geometric Constructions	370	(5)
The Fundamental Theorem of Algebra	375	(2)
An Application to Cyclic and BCH Codes	377	(16)
Finitely Generated Abelian Groups	393	(24)
Finite Abelian Groups	393	(14)
Biography of Niels Henrik Abel	404	(3)
Finitely Generated Abelian Groups	407	(10)
p-Groups and the Sylow Theorems	417	(43)
Factors and Products	418	(9)
Cauchy's Theorem and p-Groups	427	(8)
Biography of Augustin Louis Cauchy	432	(3)
Group Actions	435	(9)
The Sylow Theorems	444	(9)
Biography of Peter Ludvig Mejdell Sylow	451	(2)
An Application to Combinatorics	453	(7)
Series of Subgroups	460	(25)
The Jordan-Holder Theorem	461	(7)
Solvable Groups	468	(8)
Nilpotent Groups	476	(9)
Galois Theory	485	(40)
Galois Groups and Separability	486	(10)
The Main Theorem of Galois Theory	496	(14)
Biography of Evariste Galois	507	(3)
Insolvability of Polynomials	510	(9)
Cyclotomic Polynomials and Wedderburn's Theorem	519	(6)
Algebras	525	(20)
Finite Dimensional Algebras	525	(10)
The Wedderburn Theorems	535	(10)
Biography of Joseph Henry Maclagan Wedderburn	542	(3)
Appendix A: Complex Numbers	545	(10)
Appendix B: Matrix Arithmetic	555	(8)
Bibliography	563	(3)
Selected Answers	566	(21)
Index	587