**Pure and Applied Undergraduate Texts**

Volume: 27;
2017;
675 pp;
Hardcover

MSC: Primary 00;
Secondary 20; 13; 12

**Print ISBN: 978-1-4704-2849-5
Product Code: AMSTEXT/27**

List Price: $115.00

AMS Member Price: $92.00

MAA Member Price: $103.50

**Electronic ISBN: 978-1-4704-3661-2
Product Code: AMSTEXT/27.E**

List Price: $115.00

AMS Member Price: $92.00

MAA Member Price: $103.50

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#### Supplemental Materials

# Algebra in Action: A Course in Groups, Rings, and Fields

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*Shahriar Shahriari*

This text—based on the author's popular courses at Pomona College—provides a readable, student-friendly, and somewhat sophisticated introduction to abstract algebra. It is aimed at sophomore or junior undergraduates who are seeing the material for the first time. In addition to the usual definitions and theorems, there is ample discussion to help students build intuition and learn how to think about the abstract concepts. The book has over 1300 exercises and mini-projects of varying degrees of difficulty, and, to facilitate active learning and self-study, hints and short answers for many of the problems are provided. There are full solutions to over 100 problems in order to augment the text and to model the writing of solutions. Lattice diagrams are used throughout to visually demonstrate results and proof techniques. The book covers groups, rings, and fields. In group theory, group actions are the unifying theme and are introduced early. Ring theory is motivated by what is needed for solving Diophantine equations, and, in field theory, Galois theory and the solvability of polynomials take center stage. In each area, the text goes deep enough to demonstrate the power of abstract thinking and to convince the reader that the subject is full of unexpected results.

#### Readership

Undergraduate students interested in abstract algebra.

#### Reviews & Endorsements

Written with great care and clarity, Shahriari's "Algebra in Action" provides an excellent introduction to abstract algebra. I have used the book twice to teach abstract algebra class at Reed College, and it's a perfect fit. The book is sophisticated yet readable, and packed with examples and exercises. Group actions appear early on, serving to motivate and unify many of the important concepts in group theory. The book also includes plenty of material on rings and fields, including the basics of Galois theory.

-- Jamie Pommersheim, Reed College

The structure of the text "Algebra in Action" lets students see what groups really do right from the very beginning. In the very first chapter, the author introduces a rich selection of examples, the dihedral groups, the symmetric group, the integers modulo n, and matrix groups, that students can see 'in action' before the presentation of the formal definitions of groups and group actions in chapter 2 where the theoretical foundations are introduced. Students return to these examples again and again as the formal theory unfolds, seeing how the theory lets them study all groups at once...It is one of the few texts at the undergraduate level that supports the incorporation of group actions at an early stage in the course.

-- Jessica Sidman, Mount Holyoke College

Shahriar Shahriari has written an exquisite text that will become, and deserves to be, widely used for introducing generations of students to abstract algebra.The presentation is engaging, modern, and sufficiently detailed, making the book ideal for self-study..."Algebra in Action" is a gem and, no doubt, it is the work of a master teacher whose passion and respect for the subject is apparent everywhere in the book. I highly recommend it to students and professors alike!

-- Ehssan Khanmohammadi

It is rigorous, well-written, ample in terms of problems and solutions provided, and sufficiently advanced for its target audience.

-- Jason M. Graham, MAA Reviews

#### Table of Contents

# Table of Contents

## Algebra in Action: A Course in Groups, Rings, and Fields

- Cover Cover11
- Title page iii4
- Contents vii8
- Preface xiii14
- Part 1 . (Mostly Finite) Group Theory 120
- Chapter 1. Four Basic Examples 322
- Chapter 2. Groups: The Basics 3756
- Chapter 3. The Alternating Groups 7594
- Chapter 4. Group Actions 85104
- Chapter 5. A Subgroup Acts on the Group: Cosets and Lagrange’s Theorem 109128
- Chapter 6. A Group Acts on Itself: Counting and the Conjugation Action 129148
- Chapter 7. Acting on Subsets, Cosets, and Subgroups: The Sylow Theorems 143162
- Chapter 8. Counting the Number of Orbits^{⋆} 155174
- Chapter 9. The Lattice of Subgroups^{⋆} 167186
- Chapter 10. Acting on Its Subgroups: Normal Subgroups and Quotient Groups 187206
- Chapter 11. Group Homomorphisms 209228
- 11.1. Definitions, Examples, and Elementary Properties 210229
- 11.2. The Kernel and the Image 214233
- 11.3. Homomorphisms, Normal Subgroups, and Quotient Groups 217236
- 11.4. Actions and Homomorphisms 223242
- 11.5. The Homomorphism Theorems 227246
- 11.6. Automorphisms and Inner-automorphisms^{⋆} 238257
- 11.7. More Problems and Projects 243262

- Chapter 12. Using Sylow Theorems to Analyze Finite Groups* 249268
- Chapter 13. Direct and Semidirect Products^{⋆} 269288
- Chapter 14. Solvable and Nilpotent Groups^{⋆} 285304

- Part 2 . (Mostly Commutative) Ring Theory 307326
- Chapter 15. Rings 309328
- Chapter 16. Homomorphisms, Ideals, and Quotient Rings 327346
- Chapter 17. Field of Fractions and Localization 353372
- Chapter 18. Factorization, EDs, PIDs, and UFDs 367386
- Chapter 19. Polynomial Rings 403422
- Chapter 20. Gaussian Integers and (a little) Number Theory^{⋆} 435454

- Part 3 . Fields and Galois Theory 449468
- Chapter 21. Introducing Field Theory and Galois Theory 451470
- Chapter 22. Field Extensions 459478
- Chapter 23. Straightedge and Compass Constructions 477496
- Chapter 24. Splitting Fields and Galois Groups 491510
- Chapter 25. Galois, Normal, and Separable Extensions 515534
- Chapter 26. Fundamental Theorem of Galois Theory 547566
- Chapter 27. Finite Fields and Cyclotomic Extensions 565584
- Chapter 28. Radical Extensions, Solvable Groups, and the Quintic 589608
- Appendix A. Hints for Selected Problems 611630
- Appendix B. Short Answers for Selected Problems 619638
- Appendix C. Complete Solutions for Selected (Odd-Numbered) Problems 623642
- Bibliography 661680
- Index 665684

- Back Cover Back Cover1698