Basic Abstract Algebra For Graduate Students and Advanced Undergraduates

by Ash, Robert B.

• ISBN13: 9780486453569
• ISBN10: 0486453561
• eBook ISBN(s): 9780486318110
• Format: Paperback
• Copyright: 2006-12-01
• Publisher: Dover Publications
• More Book Details
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Summary

This survey of fundamental algebraic structures employs techniques applicable to mathematics, physics, engineering, and computer science. Topics include relations between groups and sets, the fundamental theorem of Galois theory, and the results and methods of abstract algebra in terms of number theory, geometry, and noncommutative and homological algebra. Solutions. 2006 edition.

Table of Contents

PREFACE REMARKS ON EXPOSITORY WRITING IN MATHEMATICS	ix
CHAPTER 0 PREREQUISITES
0.1 Elementary Number Theory	1	(2)
0.2 Set Theory	3	(2)
0.3 Linear Algebra	5	(2)
CHAPTER 1 GROUP FUNDAMENTALS
1.1 Groups and Subgroups	7	(4)
1.2 Permutation Groups	11	(3)
1.3 Cosets, Normal Subgroups and Homomorphisms	14	(6)
1.4 The Isomorphism Theorems	20	(5)
1.5 Direct Products	25	(4)
CHAPTER 2 RING FUNDAMENTALS
2.1 Basic Definitions and Properties	29	(4)
2.2 Ideals, Homomorphisms and Quotient Rings	33	(4)
2.3 The Isomorphism Theorems For Rings	37	(4)
2.4 Maximal and Prime Ideals	41	(3)
2.5 Polynomial Rings	44	(3)
2.6 Unique Factorization	47	(5)
2.7 Principal Ideal Domains and Euclidean Domains	52	(2)
2.8 Rings of Fractions	54	(4)
2.9 Irreducible Polynomials	58	(5)
CHAPTER 3 FIELD FUNDAMENTALS
3.1 Field Extensions	63	(4)
3.2 Splitting Fields	67	(3)
3.3 Algebraic Closures	70	(3)
3.4 Separability	73	(4)
3.5 Normal Extensions	77	(8)
CHAPTER 4 MODULE FUNDAMENTALS
4.1 Modules and Algebras	85	(4)
4.2 The Isomorphism Theorems For Modules	89	(2)
4.3 Direct Sums and Free Modules	91	(4)
4.4 Homomorphisms and Matrices	95	(2)
4.5 Smith Normal Form	97	(4)
4.6 Fundamental Structure Theorems	101	(4)
4.7 Exact Sequences and Diagram Chasing	105	(6)
ENRICHMENT	111	(12)
(Chapters 1-4)
CHAPTER 5 SOME BASIC TECHNIQUES OF GROUP THEORY
5.1 Groups Acting on Sets	123	(3)
5.2 The Orbit-Stabilizer Theorem	126	(3)
5.3 Applications to Combinatorics	129	(4)
5.4 The Sylow Theorems	133	(3)
5.5 Applications of the Sylow Theorems	136	(3)
5.6 Composition Series	139	(3)
5.7 Solvable and Nilpotent Groups	142	(4)
5.8 Generators and Relations	146	(5)
CHAPTER 6 GALOIS THEORY
6.1 Fixed Fields and Galois Groups	151	(4)
6.2 The Fundamental Theorem	155	(3)
6.3 Computing a Galois Group Directly	158	(2)
6.4 Finite Fields	160	(3)
6.5 Cyclotomic Fields	163	(4)
6.6 The Galois Group of a Cubic	167	(3)
6.7 Cyclic and Kummer Extensions	170	(3)
6.8 Solvability by Radicals	173	(5)
6.9 Transcendental Extensions	178	(3)
Appendix to Chapter 6	181	(8)
CHAPTER 7 INTRODUCING ALGEBRAIC NUMBER THEORY
(Commutative Algebra 1)	189	(1)
7.1 Integral Extensions	190	(3)
7.2 Quadratic Extensions of the Rationals	193	(2)
7.3 Norms and Traces	195	(5)
7.4 The Discriminant	200	(5)
7.5 Noetherian and Artinian Modules and Rings	205	(5)
7.6 Fractional Ideals	210	(3)
7.7 Unique Factorization of Ideals in a Dedekind Domain	213	(2)
7.8 Some Arithmetic in Dedekind Domains	215	(2)
7.9 p-adic Numbers	217	(6)
CHAPTER 8 INTRODUCING ALGEBRAIC GEOMETRY
(Commutative Algebra 2)	223	(1)
8.1 Varieties	223	(3)
8.2 The Hilbert Basis Theorem	226	(2)
8.3 The Nullstellensatz: Preliminaries	228	(5)
8.4 The Nullstellensatz: Equivalent Versions and Proof	233	(2)
8.5 Localization	235	(4)
8.6 Primary Decomposition	239	(4)
8.7 Tensor Product of Modules Over a Commutative Ring	243	(4)
8.8 General Tensor Products	247	(6)
CHAPTER 9 INTRODUCING NONCOMMUTATIVE ALGEBRA
9.1 Semisimple Modules	253	(2)
9.2 Two Key Theorems	255	(3)
9.3 Simple and Semisimple Rings	258	(4)
9.4 Further Properties of Simple Rings, Matrix Rings, and Endomorphisms	262	(3)
9.5 The Structure of Semisimple Rings	265	(5)
9.6 Maschke's Theorem	270	(3)
9.7 The Jacobson Radical	273	(3)
9.8 Theorems of Hopkins-Levitzki and Nakayama	276	(5)
CHAPTER 10 INTRODUCING HOMOLOGICAL ALGEBRA
10.1 Categories	281	(4)
10.2 Products and Coproducts	285	(3)
10.3 Functors	288	(4)
10.4 Exact Functors	292	(3)
10.5 Projective Modules	295	(3)
10.6 Injective Modules	298	(4)
10.7 Embedding into an Injective Module	302	(4)
10.8 Flat Modules	306	(3)
10.9 Direct and Inverse Limits	309	(4)
Appendix to Chapter 10	313	(4)
SUPPLEMENT
S1 Chain Complexes	317	(2)
S2 The Snake Lemma	319	(3)
S3 The Long Exact Homology Sequence	322	(2)
S4 Projective and Injective Resolutions	324	(1)
S5 Derived Functors	325	(4)
S6 Some Properties of Ext and Tor	329	(3)
S7 Base Change in the Tensor Product	332	(3)
SOLUTIONS TO PROBLEMS
CHAPTERS 1-5	335	(30)
CHAPTERS 6-10	365	(34)
BIBLIOGRAPHY	399	(2)
LIST OF SYMBOLS	401	(2)
INDEX	403