Basic Abstract Algebra For Graduate Students and Advanced Undergraduates

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ISBN: 9780486453569 | 0486453561
Cover: Paperback
Copyright: 12/1/2006

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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.

PREFACE REMARKS ON EXPOSITORY WRITING IN MATHEMATICS

CHAPTER 0 PREREQUISITES

0.1 Elementary Number Theory

(2)

0.2 Set Theory

(2)

0.3 Linear Algebra

(2)

CHAPTER 1 GROUP FUNDAMENTALS

1.1 Groups and Subgroups

(4)

1.2 Permutation Groups

(3)

1.3 Cosets, Normal Subgroups and Homomorphisms

(6)

1.4 The Isomorphism Theorems

(5)

1.5 Direct Products

(4)

CHAPTER 2 RING FUNDAMENTALS

2.1 Basic Definitions and Properties

(4)

2.2 Ideals, Homomorphisms and Quotient Rings

(4)

2.3 The Isomorphism Theorems For Rings

(4)

2.4 Maximal and Prime Ideals

(3)

2.5 Polynomial Rings

(3)

2.6 Unique Factorization

(5)

2.7 Principal Ideal Domains and Euclidean Domains

(2)

2.8 Rings of Fractions

(4)

2.9 Irreducible Polynomials

(5)

CHAPTER 3 FIELD FUNDAMENTALS

3.1 Field Extensions

(4)

3.2 Splitting Fields

(3)

3.3 Algebraic Closures

(3)

3.4 Separability

(4)

3.5 Normal Extensions

(8)

CHAPTER 4 MODULE FUNDAMENTALS

4.1 Modules and Algebras

(4)

4.2 The Isomorphism Theorems For Modules

(2)

4.3 Direct Sums and Free Modules

(4)

4.4 Homomorphisms and Matrices

(2)

4.5 Smith Normal Form

(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

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Basic Abstract Algebra For Graduate Students and Advanced Undergraduates

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