Abstract Algebra A Concrete Introduction

by Redfield, Robert H.

• ISBN13: 9780201437218
• ISBN10: 020143721X
• Edition: 1st
• Format: Paperback
• Copyright: 2000-10-18
• Publisher: Pearson

Summary

This is a new text for the Abstract Algebra course. The author has written this text with a unique, yet historical, approach: solvability by radicals. This approach depends on a fields-first organization. However, professors wishing to commence their course with group theory will find that the Table of Contents is highly flexible, and contains a generous amount of group coverage.

Table of Contents

Introduction	xvii
Historical Note: Al-Khwarizmi	xix
PART 1 PRELIMINARIES	1	(68)
Properties of the Integers	3	(15)
Historical Note: Augustus de Morgan	16	(2)
Solving Cubic and Quartic Polynomial Equations	18	(16)
Historical Note: How the Cubic and Quartic Equations Were Solved	31	(3)
Complex Numbers	34	(24)
Historical Note: Highlights in the Development of the Complex Numbers	48	(10)
Some Other Examples	58	(11)
Historical Note: William Rowan Hamilton	66	(3)
PART 2 ALGEBRAIC EXTENSION FIELDS	69	(114)
Fields	70	(11)
Solvability by Radicals	81	(14)
Historical Note: Niels Henrik Abel	91	(4)
Rings	95	(8)
Ways in Which Polynomials Are Like the Integers	103	(16)
Historical Note: Julia Robinson	116	(3)
Principal Ideals	119	(11)
Historical Note: Emmy Noether	127	(3)
Algebraic Elements	130	(10)
Eisenstein's Irreducibility Criterion	140	(8)
Historical Note: Gotthold Eisenstein	146	(2)
Extension Fields as Vector Spaces	148	(11)
Automorphisms of Fields	159	(11)
Historical Note: Evariste Galois	167	(3)
Counting Automorphisms	170	(13)
Historical Note: Richard Dedekind	180	(3)
PART 3 ELEMENTARY GROUP THEORY	183	(88)
Groups	184	(12)
Historical Note: Walther von Dyck	194	(2)
Permutation Groups	196	(7)
Group Homomorphisms	203	(8)
Historical Note: Arthur Cayley	209	(2)
Subgroups	211	(9)
Subgroups Generated by Subsets	220	(6)
Cosets	226	(7)
Finite Groups and Lagrange's Theorem	233	(10)
Historical Note: Joseph Louis Lagrange	241	(2)
Equivalence Relations and Cauchy's Theorem	243	(11)
Historical Note: Augustin-Louis Cauchy	251	(3)
Normal Subgroups and Quotient Groups	254	(8)
Historical Note: Otto Holder	259	(3)
The Homomorphism Theorem for Groups	262	(9)
Historical Note: B. L. van der Waerden	268	(3)
PART 4 POLYNOMIAL EQUATIONS NOT SOLVABLE BY RADICALS	271	(30)
Galois Groups of Radical Extensions	272	(8)
Solvable Groups and Commutator Subgroups	280	(8)
Historical Note: William Burnside	287	(1)
Solvable Galois Groups	288	(8)
Polynomial Equations Not Solvable by Radicals	296	(5)
Historical Note: Paolo Ruffini	299	(2)
PART 5 FINITE GROUPS	301	(95)
Finite External Direct Products of Groups	302	(10)
Historical Note: J. H. M. Wedderburn	310	(2)
Finite Internal Direct Products of Groups	312	(8)
Abelian Groups with Prime Power Order	320	(9)
The Fundamental Theorem of Finite Abelian Groups	329	(10)
Historical Note: Leopold Kronecker	337	(2)
Dihedral Groups	339	(9)
Historical Note: Felix Klein	345	(3)
Cauchy's Theorem	348	(11)
The Sylow Theorems	359	(13)
Historical Note: Peter Ludvig Sylow	370	(2)
Groups of Order Less Than 16	372	(8)
Groups of Even Permutations	380	(8)
Historical Note: Camille Jordan	386	(2)
Semidirect Products	388	(8)
Appendix A The Greek Alphabet	396	(1)
Appendix B Proving Theorems	397	(16)
Historical Note: George Boole	410	(3)
Appendix C Vector Spaces Over Fields	413	(5)
Appendix D Constructions with Straightedge and Compass	418	(8)
Answers to Odd-Numbered Computational Exercises	426	(18)
Bibliography	444	(2)
Photo Credits	446	(1)
Notation Index	447	(1)
Subject Index	448