**Pure and Applied Undergraduate Texts**

Volume: 55;
2022;
567 pp;
Softcover

MSC: Primary 12; 13; 16; 20;

**Print ISBN: 978-1-4704-6860-6
Product Code: AMSTEXT/55**

List Price: $49.00

AMS Member Price: $39.20

MAA Member Price: $44.10

**Electronic ISBN: 978-1-4704-6902-3
Product Code: AMSTEXT/55.E**

List Price: $45.00

AMS Member Price: $36.00

MAA Member Price: $40.50

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#### Supplemental Materials

# Abstract Algebra: An Integrated Approach

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*Joseph H. Silverman*

This abstract algebra textbook takes an integrated approach
that highlights the similarities of fundamental algebraic structures
among a number of topics. The book begins by introducing groups,
rings, vector spaces, and fields, emphasizing examples, definitions,
homomorphisms, and proofs. The goal is to explain how all of the
constructions fit into an axiomatic framework and to emphasize the
importance of studying those maps that preserve the underlying
algebraic structure. This fast-paced introduction is followed by
chapters in which each of the four main topics is revisited and deeper
results are proven.

The second half of the book contains material of a more advanced
nature. It includes a thorough development of Galois theory, a chapter
on modules, and short surveys of additional algebraic topics designed
to whet the reader's appetite for further study.

This book is intended for a first introduction to abstract algebra
and requires only a course in linear algebra as a prerequisite. The
more advanced material could be used in an introductory graduate-level
course.

An instructor's solutions manual is electronically available for
this title to those instructors who have adopted the textbook for
classroom use. Please send email to textbooks@ams.org for more
information.

#### Readership

Undergraduate and graduate students interested in abstract algebra.

#### Reviews & Endorsements

It will come as no surprise that the material is presented in a clear and flawless manner; in addition, there are many exercises and an extensive index.

-- Franz Lemmermeyer, zbMATH Open

A quick review of these archives alone will show that textbooks for undergraduate abstract algebra courses are not in short supply. Several of them are excellent, and as an instructor, I have an embarrassment of riches in choosing for my course. I expect this text will be on that list when next I get to teach the subject. Silverman's dedication says, "This one is for the next generation." Indeed, this is a wonderful resource for training the next generation of mathematicians.

-- Michele Intermont, Kalamazoo College

# Table of Contents

## Abstract Algebra: An Integrated Approach

- Preface xiii14
- Chapter 1. A Potpourri of Preliminary Topics 120
- 1.1. What Are Definitions, Axioms, and Proofs? 120
- 1.2. Mathematical Credos to Live By! 221
- 1.3. A Smidgeon of Mathematical Logic and Some Proof Techniques 322
- 1.4. A Smidgeon of Set Theory 928
- 1.5. Functions 1231
- 1.6. Equivalence Relations 1332
- 1.7. Mathematical Induction 1635
- 1.8. A Smidgeon of Number Theory 1736
- 1.9. A Smidgeon of Combinatorics 2140
- Exercises 2645

- Chapter 2. Groups — Part 1 3554
- Chapter 3. Rings — Part 1 6382
- 3.1. Introduction to Rings 6382
- 3.2. Abstract Rings and Ring Homomorphisms 6382
- 3.3. Interesting Examples of Rings 6584
- 3.4. Some Important Special Types of Rings 6988
- 3.5. Unit Groups and Product Rings 7089
- 3.6. Ideals and Quotient Rings 7291
- 3.7. Prime Ideals and Maximal Ideals 7796
- Exercises 7998

- Chapter 4. Vector Spaces — Part 1 91110
- Chapter 5. Fields — Part 1 105124
- Chapter 6. Groups — Part 2 127146
- Chapter 7. Rings — Part 2 157176
- 7.1. Irreducible Elements and Unique Factorization Domains 157176
- 7.2. Euclidean Domains and Principal Ideal Domains 159178
- 7.3. Factorization in Principal Ideal Domains 164183
- 7.4. The Chinese Remainder Theorem 167186
- 7.5. Field of Fractions 172191
- 7.6. Multivariate and Symmetric Polynomials 176195
- Exercises 180199

- Chapter 8. Fields — Part 2 187206
- 8.1. Algebraic Numbers and Transcendental Numbers 187206
- 8.2. Polynomial Roots and Multiplicative Subgroups 190209
- 8.3. Splitting Fields, Separability, and Irreducibility 194213
- 8.4. Finite Fields Revisited 200219
- 8.5. Gauss's Lemma and Eisenstein's Irreducibility Criterion 201220
- 8.6. Ruler and Compass Constructions 207226
- Exercises 214233

- Chapter 9. Galois Theory: Fields+Groups 221240
- 9.1. What Is Galois Theory? 221240
- 9.2. A Quick Review of Polynomials and Field Extensions 222241
- 9.3. Fields of Algebraic Numbers 223242
- 9.4. Algebraically Closed Fields 226245
- 9.5. Automorphisms of Fields 227246
- 9.6. Splitting Fields — Part 1 229248
- 9.7. Splitting Fields — Part 2 235254
- 9.8. The Primitive Element Theorem 239258
- 9.9. Galois Extensions 242261
- 9.10. The Fundamental Theorem of Galois Theory 247266
- 9.11. Application: The Fundamental Theorem of Algebra 250269
- 9.12. Galois Theory of Finite Fields 254273
- 9.13. A Plethora of Galois Equivalences 257276
- 9.14. Cyclotomic Fields and Kummer Fields 264283
- 9.15. Application: Insolubility of Polynomial Equations by Radicals 269288
- 9.16. Linear Independence of Field Automorphisms 281300
- Exercises 285304

- Chapter 10. Vector Spaces — Part 2 295314
- 10.1. Vector Space Homomorphisms (aka Linear Transformations) 295314
- 10.2. Endomorphisms and Automorphisms 296315
- 10.3. Linear Transformations and Matrices 298317
- 10.4. Subspaces and Quotient Spaces 303322
- 10.5. Eigenvalues and Eigenvectors 306325
- 10.6. Determinants 309328
- 10.7. Determinants, Eigenvalues, and Characteristic Polynomials 315334
- 10.8. Inifinite-Dimensional Vector Spaces 318337
- Exercises 320339

- Chapter 11. Modules — Part 1:Rings+Vector-Like Spaces 327346
- 11.1. What Is a Module? 327346
- 11.2. Examples of Modules 328347
- 11.3. Submodules and Quotient Modules 330349
- 11.4. Free Modules and Finitely Generated Modules 332351
- 11.5. Homomorphisms, Endomorphisms, Matrices 334353
- 11.6. Noetherian Rings and Modules 337356
- 11.7. Matrices with Entries in a Euclidean Domain 343362
- 11.8. Finitely Generated Modules over Euclidean Domains 346365
- 11.9. Applications of the Structure Theorem 353372
- Exercises 357376

- Chapter 12. Groups — Part 3 371390
- Chapter 13. Modules — Part 2: Multilinear Algebra 397416
- Chapter 14. Additional Topics in Brief 413432
- 14.1. Sets Countable and Uncountable 413432
- 14.2. The Axiom of Choice 417436
- 14.3. Tensor Products and Multilinear Algebra 422441
- 14.4. Commutative Algebra 426445
- 14.5. Category Theory 435454
- 14.6. Graph Theory 443462
- 14.7. Representation Theory 448467
- 14.8. Elliptic Curves 457476
- 14.9. Algebraic Number Theory 464483
- 14.10. Algebraic Geometry 470489
- 14.11. Euclidean Lattices 477496
- 14.12. Non-Commutative Rings 489508
- 14.13. Mathematical Cryptography 496515
- Exercises 504523

- Sample Syllabi 523542
- List of Notation 527546
- List of Figures 533552
- Index 537556