**AMS/MAA Textbooks**

Volume: 65;
2021;
478 pp;
Softcover

MSC: Primary 20; 16; 12; 06;

**Print ISBN: 978-1-4704-6030-3
Product Code: TEXT/65**

List Price: $85.00

AMS Member Price: $63.75

MAA Member Price: $63.75

**Electronic ISBN: 978-1-4704-6306-9
Product Code: TEXT/65.E**

List Price: $85.00

AMS Member Price: $63.75

MAA Member Price: $63.75

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#### Supplemental Materials

# Thinking Algebraically: An Introduction to Abstract Algebra

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*Thomas Q. Sibley*

MAA Press: An Imprint of the American Mathematical Society

Thinking Algebraically presents the insights of abstract
algebra in a welcoming and accessible way. It succeeds in combining
the advantages of rings-first and groups-first approaches while
avoiding the disadvantages. After an historical overview, the first
chapter studies familiar examples and elementary properties of groups
and rings simultaneously to motivate the modern understanding of
algebra. The text builds intuition for abstract algebra starting from
high school algebra. In addition to the standard number systems,
polynomials, vectors, and matrices, the first chapter introduces
modular arithmetic and dihedral groups. The second chapter builds on
these basic examples and properties, enabling students to learn
structural ideas common to rings and groups: isomorphism,
homomorphism, and direct product. The third chapter investigates
introductory group theory. Later chapters delve more deeply into
groups, rings, and fields, including Galois theory, and they also
introduce other topics, such as lattices. The exposition is clear and
conversational throughout.

The book has numerous exercises in each section as well as
supplemental exercises and projects for each chapter. Many examples
and well over 100 figures provide support for learning. Short
biographies introduce the mathematicians who proved many of the
results. The book presents a pathway to algebraic thinking in a
semester- or year-long algebra course.

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

#### Readership

Undergraduate students interested in abstract algebra.

#### Table of Contents

# Table of Contents

## Thinking Algebraically: An Introduction to Abstract Algebra

- Cover Cover11
- Title page iii5
- Copyright iv6
- Contents v7
- Preface ix11
- Prologue 117
- Chapter 1. A Transitionto Abstract Algebra 319
- Chapter 2. Relationshipsbetween Systems 5167
- Chapter 3. Groups 99115
- 3.1. Cyclic Groups 99115
- Exercises 103119
- 3.2. Abelian Groups 108124
- Exercises 113129
- 3.3. Cayley Digraphs 120136
- Exercises 124140
- 3.4. Group Actions and Finite Symmetry Groups 127143
- Exercises 134150
- 3.5. Permutation Groups, Part I 140156
- Exercises 145161
- 3.6. Normal Subgroups and Factor Groups 150166
- Exercises 158174
- 3.7. Permutation Groups, Part II 162178
- Exercises 165181
- Supplemental Exercises 169185
- Projects 172188
- Appendix: The Fundamental Theorem* of Finite Abelian Groups 177193

- Chapter 4. Rings, Integral Domains,and Fields 181197
- 4.1. Rings and Integral Domains 181197
- Exercises 187203
- 4.2. Ideals and Factor Rings 190206
- Exercises 194210
- 4.3. Prime and Maximal Ideals 197213
- Exercises 203219
- 4.4. Properties of Integral Domains 207223
- Exercises 215231
- 4.5. Gröbner Bases in Algebraic Geometry 219235
- Exercises 225241
- 4.6. Polynomial Dynamical Systems 228244
- Exercises 232248
- Supplemental Exercises 234250
- Projects 236252

- Chapter 5. Vector Spacesand Field Extensions 243259
- 5.1. Vector Spaces 244260
- Exercises 250266
- 5.2. Linear Codes and Cryptography 255271
- Exercises 261277
- 5.3. Algebraic Extensions 266282
- Exercises 273289
- 5.4. Geometric Constructions 277293
- Exercises 286302
- 5.5. Splitting Fields 290306
- Exercises 297313
- 5.6. Automorphisms of Fields 302318
- Exercises 308324
- 5.7. Galois Theory* and the Insolvability of the Quintic 312328
- Exercises 319335
- Supplemental Exercises 322338
- Projects 325341

- Chapter 6. Topics in Group Theory 327343
- 6.1. Finite Symmetry Groups 327343
- Exercises 335351
- 6.2. Frieze, Wallpaper, and Crystal Patterns 341357
- Exercises 351367
- 6.3. Matrix Groups 356372
- Exercises 364380
- 6.4. Semidirect Products of Groups 370386
- Exercises 376392
- 6.5. The Sylow Theorems 381397
- Exercises 388404
- Supplemental Exercises 391407
- Projects 395411

- Chapter 7. Topics in Algebra 399415
- Epilogue 439455
- Selected Answers 443459
- Terms 469485
- Symbols 475491
- Names 477493
- Back Cover Back Cover1496