A First Course in Abstract Algebra

by Fraleigh, John B.

• ISBN13: 9780201763904
• ISBN10: 0201763907
• Edition: 7th
• Format: Paperback
• Copyright: 2002-11-06
• Publisher: Pearson

Summary

Considered a classic by many,A First Course in Abstract Algebra, Seventh Edition is an in-depth introduction to abstract algebra. Focused on groups, rings and fields, this text gives students a firm foundation for more specialized work by emphasizing an understanding of the nature of algebraic structures. Sets and Relations; GROUPS AND SUBGROUPS; Introduction and Examples; Binary Operations; Isomorphic Binary Structures; Groups; Subgroups; Cyclic Groups; Generators and Cayley Digraphs; PERMUTATIONS, COSETS, AND DIRECT PRODUCTS; Groups of Permutations; Orbits, Cycles, and the Alternating Groups; Cosets and the Theorem of Lagrange; Direct Products and Finitely Generated Abelian Groups; Plane Isometries; HOMOMORPHISMS AND FACTOR GROUPS; Homomorphisms; Factor Groups; Factor-Group Computations and Simple Groups; Group Action on a Set; Applications of G-Sets to Counting; RINGS AND FIELDS; Rings and Fields; Integral Domains; Fermat's and Euler's Theorems; The Field of Quotients of an Integral Domain; Rings of Polynomials; Factorization of Polynomials over a Field; Noncommutative Examples; Ordered Rings and Fields; IDEALS AND FACTOR RINGS; Homomorphisms and Factor Rings; Prime and Maximal Ideas; Grouml;bner Bases for Ideals; EXTENSION FIELDS; Introduction to Extension Fields; Vector Spaces; Algebraic Extensions; Geometric Constructions; Finite Fields; ADVANCED GROUP THEORY; Isomorphism Theorems; Series of Groups; Sylow Theorems; Applications of the Sylow Theory; Free Abelian Groups; Free Groups; Group Presentations; GROUPS IN TOPOLOGY; Simplicial Complexes and Homology Groups; Computations of Homology Groups; More Homology Computations and Applications; Homological Algebra; Factorization; Unique Factorization Domains; Euclidean Domains; Gaussian Integers and Multiplicative Norms; AUTOMORPHISMS AND GALOIS THEORY; Automorphisms of Fields; The Isomorphism Extension Theorem; Splitting Fields; Separable Extensions; Totally Inseparable Extensions; Galois Theory; Illustrations of Galois Theory; Cyclotomic Extensions; Insolvability of the Quintic; Matrix Algebra For all readers interested in abstract algebra.

Table of Contents

Instructor's Preface	vii
Student's Preface	xi
Dependence Chart	xiii
Sets and Relations	1	(10)
I Groups and Subgroups	11	(64)
Introduction and Examples	11	(9)
Binary Operations	20	(8)
Isomorphic Binary Structures	28	(8)
Groups	36	(13)
Subgroups	49	(10)
Cyclic Groups	59	(9)
Generating Sets and Cayley Digraphs	68	(7)
II Permutations, Cosets, and Direct Products	75	(50)
Groups of Permutations	75	(12)
Orbits, Cycles, and the Alternating Groups	87	(9)
Cosets and the Theorem of Lagrange	96	(8)
Direct Products and Finitely Generated Abelian Groups	104	(10)
Plane Isometries	114	(11)
III Homomorphisms and Factor Groups	125	(42)
Homomorphisms	125	(10)
Factor Groups	135	(9)
Factor-Group Computations and Simple Groups	144	(10)
Group Action on a Set	154	(7)
Applications of G-Sets to Counting	161	(6)
IV Rings and Fields	167	(70)
Rings and Fields	167	(10)
Integral Domains	177	(7)
Fermat's and Euler's Theorems	184	(6)
The Field of Quotients of an Integral Domain	190	(8)
Rings of Polynomials	198	(11)
Factorization of Polynomials over a Field	209	(11)
Noncommutative Examples	220	(7)
Ordered Rings and Fields	227	(10)
V Ideals and Factor Rings	237	(28)
Homomorphisms and Factor Rings	237	(8)
Prime and Maximal Ideals	245	(9)
Grobner Bases for Ideals	254	(11)
VI Extension Fields	265	(42)
Introduction to Extension Fields	265	(9)
Vector Spaces	274	(9)
Algebraic Extensions	283	(10)
Geometric Constructions	293	(7)
Finite Fields	300	(7)
VII Advanced Group Theory	307	(48)
Isomorphism Theorems	307	(4)
Series of Groups	311	(10)
Sylow Theorems	321	(6)
Applications of the Sylow Theory	327	(6)
Free Abelian Groups	333	(8)
Free Groups	341	(5)
Group Presentations	346	(9)
VIII Groups in Topology	355	(34)
Simplicial Complexes and Homology Groups	355	(8)
Computations of Homology Groups	363	(8)
More Homology Computations and Applications	371	(8)
Homological Algebra	379	(10)
IX Factorization	389	(26)
Unique Factorization Domains	389	(12)
Euclidean Domains	401	(6)
Gaussian Integers and Multiplicative Norms	407	(8)
X Automorphisms and Galois Theory	415	(62)
Automorphisms of Fields	415	(9)
The Isomorphism Extension Theorem	424	(7)
Splitting Fields	431	(5)
Separable Extensions	436	(8)
Totally Inseparable Extensions	444	(4)
Galois Theory	448	(9)
Illustrations of Galois Theory	457	(7)
Cyclotomic Extensions	464	(6)
Insolvability of the Quintic	470	(7)
Appendix: Matrix Algebra	477	(6)
Bibliography	483	(4)
Notations	487	(4)
Answers to Odd-Numbered Exercises Not Asking for Definitions or Proofs	491	(22)
Index	513