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Hardcover ISBN:  9781470428495 
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Hardcover ISBN:  9781470428495 
Product Code:  AMSTEXT/27 
List Price:  $99.00 
MAA Member Price:  $89.10 
AMS Member Price:  $79.20 
eBook ISBN:  9781470436612 
Product Code:  AMSTEXT/27.E 
List Price:  $85.00 
MAA Member Price:  $76.50 
AMS Member Price:  $68.00 
Hardcover ISBN:  9781470428495 
eBook ISBN:  9781470436612 
Product Code:  AMSTEXT/27.B 
List Price:  $184.00 $141.50 
MAA Member Price:  $165.60 $127.35 
AMS Member Price:  $147.20 $113.20 

Book DetailsPure and Applied Undergraduate TextsVolume: 27; 2017; 675 ppMSC: Primary 00; Secondary 20; 13; 12;
This text—based on the author's popular courses at Pomona College—provides a readable, studentfriendly, 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 miniprojects of varying degrees of difficulty, and, to facilitate active learning and selfstudy, 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.
ReadershipUndergraduate students interested in abstract algebra.

Table of Contents

Cover

Title page

Contents

Preface

Part 1 . (Mostly Finite) Group Theory

Chapter 1. Four Basic Examples

1.1. Symmetries of a Square

1.2. 11 and Onto Functions

1.3. Integers \bmod𝑛 and Elementary Properties of Integers

1.4. Invertible Matrices

1.5. More Problems and Projects

Chapter 2. Groups: The Basics

2.1. Definitions and Examples

2.2. Cancellation Properties

2.3. Cyclic Groups and the Order of an Element

2.4. Isomorphisms

2.5. Direct Products (New Groups from Old Groups)

2.6. Subgroups

2.7. More Problems and Projects

Chapter 3. The Alternating Groups

3.1. Permutations, Cycles, and Transpositions

3.2. Even and Odd Permutations and 𝐴_{𝑛}

3.3. More Problems and Projects

Chapter 4. Group Actions

4.1. Definition and Examples

4.2. The Cayley Graph of a Group Action*

4.3. Stabilizers

4.4. Orbits

4.5. More Problems and Projects

Chapter 5. A Subgroup Acts on the Group: Cosets and Lagrange’s Theorem

5.1. Translation Action and Cosets

5.2. Lagrange’s Theorem

5.3. Application to Number Theory^{⋆}

5.4. More Problems and Projects

Chapter 6. A Group Acts on Itself: Counting and the Conjugation Action

6.1. The Fundamental Counting Principle

6.2. The Conjugation Action

6.3. The Class Equation and Groups of Order 𝑝²

6.4. More Problems and Projects

Chapter 7. Acting on Subsets, Cosets, and Subgroups: The Sylow Theorems

7.1. Binomial Coefficients \bmod𝑝

7.2. The Sylow E(xistence) Theorem

7.3. The Number and Conjugacy of Sylow Subgroups^{⋆}

Chapter 8. Counting the Number of Orbits^{⋆}

8.1. The Cauchy–Frobenius Counting Lemma

8.2. Combinatorial Applications of the Counting Lemma

8.3. More Problems and Projects

Chapter 9. The Lattice of Subgroups^{⋆}

9.1. Partially Ordered Sets, Hasse Diagrams, and Lattices

9.2. Edge Lengths and Partial Lattice Diagrams

9.3. More Problems and Projects

Chapter 10. Acting on Its Subgroups: Normal Subgroups and Quotient Groups

10.1. Normal Subgroups

10.2. The Normalizer

10.3. Quotient Groups

10.4. More Problems and Projects

Chapter 11. Group Homomorphisms

11.1. Definitions, Examples, and Elementary Properties

11.2. The Kernel and the Image

11.3. Homomorphisms, Normal Subgroups, and Quotient Groups

11.4. Actions and Homomorphisms

11.5. The Homomorphism Theorems

11.6. Automorphisms and Innerautomorphisms^{⋆}

11.7. More Problems and Projects

Chapter 12. Using Sylow Theorems to Analyze Finite Groups*

12.1. 𝑝groups

12.2. Acting on Cosets and Existence of Normal Subgroups

12.3. Applying the Sylow Theorems

12.4. 𝐴₅ Is the Only Simple Group of Order 60

Chapter 13. Direct and Semidirect Products^{⋆}

13.1. Direct Products of Groups

13.2. Fundamental Theorem of Finite Abelian Groups

13.3. Semidirect Products

13.4. Groups of Very Small Order

Chapter 14. Solvable and Nilpotent Groups^{⋆}

14.1. Solvable Groups

14.2. Nilpotent Groups

14.3. The Jordan–Hölder Theorem

Part 2 . (Mostly Commutative) Ring Theory

Chapter 15. Rings

15.1. Diophantine Equations and Rings

15.2. Rings, Integral Domains, Division Rings, and Fields

15.3. Finite Integral Domains

Chapter 16. Homomorphisms, Ideals, and Quotient Rings

16.1. Subrings, Homomorphisms, and Ideals

16.2. Quotient Rings and Homomorphism Theorems

16.3. Characteristic of Rings with Identity, Integral Domains, and Fields

16.4. Manipulating Ideals^{⋆}

Chapter 17. Field of Fractions and Localization

17.1. Field of Fractions and Localization of an Integral Domain

17.2. Localization of Commutative Rings with Identity^{⋆}

Chapter 18. Factorization, EDs, PIDs, and UFDs

18.1. Factorization in Commutative Rings

18.2. Ascending Chain Condition and Noetherian Rings

18.3. A PID is a UFD

18.4. Euclidean Domains

18.5. The Greatest Common Divisor^{⋆}

18.6. More Problems and Projects

Chapter 19. Polynomial Rings

19.1. Polynomials

19.2. 𝐾 a field ⇒ 𝐾[𝑥] an ED

19.3. Roots of Polynomials and Construction of Finite Fields

19.4. 𝑅 UFD ⇒ 𝑅[𝑥] UFD and Gauss’s Lemma

19.5. Irreducibility Criteria

19.6. Hilbert Basis Theorem^{⋆}

19.7. More Problems and Projects

Chapter 20. Gaussian Integers and (a little) Number Theory^{⋆}

20.1. Gaussian Integers

20.2. Unique Factorization and Diophantine Equations

Part 3 . Fields and Galois Theory

Chapter 21. Introducing Field Theory and Galois Theory

21.1. The Classical Problems of Field Theory

21.2. Roots of Equations, Fields, and Groups—An Example

21.3. A Quick Review of Ring Theory

Chapter 22. Field Extensions

22.1. Simple and Algebraic Extensions

22.2. A Quick Review of Vector Spaces

22.3. The Degree of an Extension

Chapter 23. Straightedge and Compass Constructions

23.1. The Field of Constructible Numbers

23.2. Characterizing Constructible Numbers

Chapter 24. Splitting Fields and Galois Groups

24.1. Roots of Polynomials, Field Extensions, and 𝐹isomorphisms

24.2. Splitting Fields

24.3. Galois Groups and Their Actions on Roots

Chapter 25. Galois, Normal, and Separable Extensions

25.1. Subgroups of the Galois Group and Intermediate Fields

25.2. Galois, Normal, and Separable Extensions

25.3. More on Normal Extensions

25.4. More on Separable Extensions

25.5. Simple Extensions

25.6. More Problems and Projects

Chapter 26. Fundamental Theorem of Galois Theory

26.1. Galois Groups and Fixed Fields

26.2. Fundamental Theorem of Galois Theory

26.3. Examples of Galois Groups

Chapter 27. Finite Fields and Cyclotomic Extensions

27.1. Finite Fields

27.2. Cyclotomic Extensions

27.3. The Polynomial 𝑥ⁿ𝑎

27.4. More Problems and Projects

Chapter 28. Radical Extensions, Solvable Groups, and the Quintic

28.1. Solvability by Radicals

28.2. A Solvable Polynomial Has a Solvable Galois Group

28.3. A Solvable Galois Group Corresponds to a Solvable Polynomial^{⋆}

28.4. More Problems and Projects

Appendix A. Hints for Selected Problems

Appendix B. Short Answers for Selected Problems

Appendix C. Complete Solutions for Selected (OddNumbered) Problems

Bibliography

Index

Back Cover


Additional Material

Reviews

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 selfstudy..."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, wellwritten, ample in terms of problems and solutions provided, and sufficiently advanced for its target audience.
Jason M. Graham, MAA Reviews


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This text—based on the author's popular courses at Pomona College—provides a readable, studentfriendly, 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 miniprojects of varying degrees of difficulty, and, to facilitate active learning and selfstudy, 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.
Undergraduate students interested in abstract algebra.

Cover

Title page

Contents

Preface

Part 1 . (Mostly Finite) Group Theory

Chapter 1. Four Basic Examples

1.1. Symmetries of a Square

1.2. 11 and Onto Functions

1.3. Integers \bmod𝑛 and Elementary Properties of Integers

1.4. Invertible Matrices

1.5. More Problems and Projects

Chapter 2. Groups: The Basics

2.1. Definitions and Examples

2.2. Cancellation Properties

2.3. Cyclic Groups and the Order of an Element

2.4. Isomorphisms

2.5. Direct Products (New Groups from Old Groups)

2.6. Subgroups

2.7. More Problems and Projects

Chapter 3. The Alternating Groups

3.1. Permutations, Cycles, and Transpositions

3.2. Even and Odd Permutations and 𝐴_{𝑛}

3.3. More Problems and Projects

Chapter 4. Group Actions

4.1. Definition and Examples

4.2. The Cayley Graph of a Group Action*

4.3. Stabilizers

4.4. Orbits

4.5. More Problems and Projects

Chapter 5. A Subgroup Acts on the Group: Cosets and Lagrange’s Theorem

5.1. Translation Action and Cosets

5.2. Lagrange’s Theorem

5.3. Application to Number Theory^{⋆}

5.4. More Problems and Projects

Chapter 6. A Group Acts on Itself: Counting and the Conjugation Action

6.1. The Fundamental Counting Principle

6.2. The Conjugation Action

6.3. The Class Equation and Groups of Order 𝑝²

6.4. More Problems and Projects

Chapter 7. Acting on Subsets, Cosets, and Subgroups: The Sylow Theorems

7.1. Binomial Coefficients \bmod𝑝

7.2. The Sylow E(xistence) Theorem

7.3. The Number and Conjugacy of Sylow Subgroups^{⋆}

Chapter 8. Counting the Number of Orbits^{⋆}

8.1. The Cauchy–Frobenius Counting Lemma

8.2. Combinatorial Applications of the Counting Lemma

8.3. More Problems and Projects

Chapter 9. The Lattice of Subgroups^{⋆}

9.1. Partially Ordered Sets, Hasse Diagrams, and Lattices

9.2. Edge Lengths and Partial Lattice Diagrams

9.3. More Problems and Projects

Chapter 10. Acting on Its Subgroups: Normal Subgroups and Quotient Groups

10.1. Normal Subgroups

10.2. The Normalizer

10.3. Quotient Groups

10.4. More Problems and Projects

Chapter 11. Group Homomorphisms

11.1. Definitions, Examples, and Elementary Properties

11.2. The Kernel and the Image

11.3. Homomorphisms, Normal Subgroups, and Quotient Groups

11.4. Actions and Homomorphisms

11.5. The Homomorphism Theorems

11.6. Automorphisms and Innerautomorphisms^{⋆}

11.7. More Problems and Projects

Chapter 12. Using Sylow Theorems to Analyze Finite Groups*

12.1. 𝑝groups

12.2. Acting on Cosets and Existence of Normal Subgroups

12.3. Applying the Sylow Theorems

12.4. 𝐴₅ Is the Only Simple Group of Order 60

Chapter 13. Direct and Semidirect Products^{⋆}

13.1. Direct Products of Groups

13.2. Fundamental Theorem of Finite Abelian Groups

13.3. Semidirect Products

13.4. Groups of Very Small Order

Chapter 14. Solvable and Nilpotent Groups^{⋆}

14.1. Solvable Groups

14.2. Nilpotent Groups

14.3. The Jordan–Hölder Theorem

Part 2 . (Mostly Commutative) Ring Theory

Chapter 15. Rings

15.1. Diophantine Equations and Rings

15.2. Rings, Integral Domains, Division Rings, and Fields

15.3. Finite Integral Domains

Chapter 16. Homomorphisms, Ideals, and Quotient Rings

16.1. Subrings, Homomorphisms, and Ideals

16.2. Quotient Rings and Homomorphism Theorems

16.3. Characteristic of Rings with Identity, Integral Domains, and Fields

16.4. Manipulating Ideals^{⋆}

Chapter 17. Field of Fractions and Localization

17.1. Field of Fractions and Localization of an Integral Domain

17.2. Localization of Commutative Rings with Identity^{⋆}

Chapter 18. Factorization, EDs, PIDs, and UFDs

18.1. Factorization in Commutative Rings

18.2. Ascending Chain Condition and Noetherian Rings

18.3. A PID is a UFD

18.4. Euclidean Domains

18.5. The Greatest Common Divisor^{⋆}

18.6. More Problems and Projects

Chapter 19. Polynomial Rings

19.1. Polynomials

19.2. 𝐾 a field ⇒ 𝐾[𝑥] an ED

19.3. Roots of Polynomials and Construction of Finite Fields

19.4. 𝑅 UFD ⇒ 𝑅[𝑥] UFD and Gauss’s Lemma

19.5. Irreducibility Criteria

19.6. Hilbert Basis Theorem^{⋆}

19.7. More Problems and Projects

Chapter 20. Gaussian Integers and (a little) Number Theory^{⋆}

20.1. Gaussian Integers

20.2. Unique Factorization and Diophantine Equations

Part 3 . Fields and Galois Theory

Chapter 21. Introducing Field Theory and Galois Theory

21.1. The Classical Problems of Field Theory

21.2. Roots of Equations, Fields, and Groups—An Example

21.3. A Quick Review of Ring Theory

Chapter 22. Field Extensions

22.1. Simple and Algebraic Extensions

22.2. A Quick Review of Vector Spaces

22.3. The Degree of an Extension

Chapter 23. Straightedge and Compass Constructions

23.1. The Field of Constructible Numbers

23.2. Characterizing Constructible Numbers

Chapter 24. Splitting Fields and Galois Groups

24.1. Roots of Polynomials, Field Extensions, and 𝐹isomorphisms

24.2. Splitting Fields

24.3. Galois Groups and Their Actions on Roots

Chapter 25. Galois, Normal, and Separable Extensions

25.1. Subgroups of the Galois Group and Intermediate Fields

25.2. Galois, Normal, and Separable Extensions

25.3. More on Normal Extensions

25.4. More on Separable Extensions

25.5. Simple Extensions

25.6. More Problems and Projects

Chapter 26. Fundamental Theorem of Galois Theory

26.1. Galois Groups and Fixed Fields

26.2. Fundamental Theorem of Galois Theory

26.3. Examples of Galois Groups

Chapter 27. Finite Fields and Cyclotomic Extensions

27.1. Finite Fields

27.2. Cyclotomic Extensions

27.3. The Polynomial 𝑥ⁿ𝑎

27.4. More Problems and Projects

Chapter 28. Radical Extensions, Solvable Groups, and the Quintic

28.1. Solvability by Radicals

28.2. A Solvable Polynomial Has a Solvable Galois Group

28.3. A Solvable Galois Group Corresponds to a Solvable Polynomial^{⋆}

28.4. More Problems and Projects

Appendix A. Hints for Selected Problems

Appendix B. Short Answers for Selected Problems

Appendix C. Complete Solutions for Selected (OddNumbered) Problems

Bibliography

Index

Back Cover

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 selfstudy..."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, wellwritten, ample in terms of problems and solutions provided, and sufficiently advanced for its target audience.
Jason M. Graham, MAA Reviews