Introduction To Abstract Algebra, An: Sets, Groups, Rings, And Fields

by Steven Howard Weintraub

Overview

This book is a textbook for a semester-long or year-long introductory course in abstract algebra at the upper undergraduate or beginning graduate level.It treats set theory, group theory, ring and ideal theory, and field theory (including Galois theory), and culminates with a treatment of Dedekind rings, including rings of algebraic integers.In addition to treating standard topics, it contains material not often dealt with in books at this level. It provides a fresh perspective on the subjects it covers, with, in particular, distinctive treatments of factorization theory in integral domains and of Galois theory.As an introduction, it presupposes no prior knowledge of abstract algebra, but provides a well-motivated, clear, and rigorous treatment of the subject, illustrated by many examples. Written with an eye toward number theory, it contains numerous applications to number theory (including proofs of Fermat's theorem on sums of two squares and of the Law of Quadratic Reciprocity) and serves as an excellent basis for further study in algebra in general and number theory in particular.Each of its chapters concludes with a variety of exercises ranging from the straightforward to the challenging in order to reinforce students' knowledge of the subject. Some of these are particular examples that illustrate the theory while others are general results that develop the theory further.

Product Details

ISBN-13:	9789811247552
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	06/21/2022
Pages:	438
Product dimensions:	6.00(w) x 9.00(h) x 0.89(d)

Table of Contents

Preface vii

About the Author xiii

1 Set Theory 1

1.1 Basic set theory 1

1.2 Relations, functions, and equivalence relations 6

1.3 Cardinality 13

1.4 Naïve and not-so-naïve set theory 23

1.5 Exercises 25

2 Group Theory 29

2.1 Definition, examples, and basic properties 29

2.2 Homomorphisms and isomorphisms 39

2.3 Subgroups 45

2.4 Quotient groups 54

2.5 The Noether isomorphism theorems 65

2.6 The structure of finite, and finitely generated, Abelian groups 68

2.7 Applications to number theory 95

2.8 Actions of groups on sets 103

2.9 Structural results on Nonabelian groups 109

2.10 The symmetric groups 123

2.11 Exercises 138

3 Ring Theory 149

3.1 Definition, examples, and basic properties 149

3.2 Ideals in rings 160

3.3 The integers, and rings of polynomials 172

3.4 Euclidean domains and principal ideal domains 178

3.5 Integral domains and divisibility 184

3.6 Principal ideal domains and unique factorization domains 191

3.7 Euclid's algorithm 201

3.8 Applications to number theory 214

3.9 Some examples of integral domains 225

3.10 Quotient fields and localization 230

3.11 Polynomial rings: Unique factorization and related matters 233

3.12 Ideals: Maximal and prime 240

3.13 Exercises 246

4 Field Theory 265

4.1 Definition, examples, and basic properties 265

4.2 Extension fields 270

4.3 Finite extensions 278

4.4 Algebraic elements 284

4.5 Adjoining elements 292

4.6 Examples of field extensions 297

4.7 Isomorphisms of fields 305

4.8 The Galois group: Definition and examples 309

4.9 Normal, separable, and Galois extensions 314

4.10 The fundamental theorem of Galois theory 320

4.11 More on Galois groups 330

4.12 Simple extensions 337

4.13 Finite fields 346

4.14 Cyclotomic fields 347

4.15 Solvability and unsolvability of equations 351

4.16 Straightedge and compass constructions 354

4.17 The fundamental theorem of algebra 361

4.18 Exercises 362

5 Rings of Algebraic Integers and Dedekind Rings 375

5.1 Rings of algebraic integers 375

5.2 Dedekind rings 381

5.3 Ideals in Dedekind rings 385

5.4 Examples 391

5.5 Further developments 401

5.6 Exercises 402

Appendix A Some Properties of the Integers 409

Appendix B A Theorem from Linear Algebra 413

Index 415