Item Successfully Added to Cart
An error was encountered while trying to add the item to the cart. Please try again.
OK
Please make all selections above before adding to cart
OK
Share this page via the icons above, or by copying the link below:
Copy To Clipboard
Successfully Copied!
Algebra in Action: A Course in Groups, Rings, and Fields
 
Shahriar Shahriari Pomona College, Claremont, CA
Front Cover for Algebra in Action
HardcoverISBN:  978-1-4704-2849-5
Product Code:  AMSTEXT/27
List Price: $115.00
MAA Member Price: $103.50
AMS Member Price: $92.00
eBookISBN:  978-1-4704-3661-2
Product Code:  AMSTEXT/27.E
List Price: $115.00
MAA Member Price: $103.50
AMS Member Price: $92.00
HardcoverISBN:  978-1-4704-2849-5
eBookISBN:  978-1-4704-3661-2
Product Code:  AMSTEXT/27.B
List Price: $230.00$172.50
MAA Member Price: $207.00$155.25
AMS Member Price: $184.00$138.00
Front Cover for Algebra in Action
Click above image for expanded view
  • Front Cover for Algebra in Action
  • Back Cover for Algebra in Action
Algebra in Action: A Course in Groups, Rings, and Fields
Shahriar Shahriari Pomona College, Claremont, CA
Hardcover ISBN:  978-1-4704-2849-5
Product Code:  AMSTEXT/27
List Price: $115.00
MAA Member Price: $103.50
AMS Member Price: $92.00
eBook ISBN:  978-1-4704-3661-2
Product Code:  AMSTEXT/27.E
List Price: $115.00
MAA Member Price: $103.50
AMS Member Price: $92.00
Hardcover ISBN:  978-1-4704-2849-5
eBookISBN:  978-1-4704-3661-2
Product Code:  AMSTEXT/27.B
List Price: $230.00$172.50
MAA Member Price: $207.00$155.25
AMS Member Price: $184.00$138.00
  • Book Details
     
     
    Pure and Applied Undergraduate Texts
    Volume: 272017; 675 pp
    MSC: Primary 00; Secondary 20; 13; 12;

    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.

  • 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. 1-1 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 Inner-automorphisms^{⋆}
    • 11.7. More Problems and Projects
    • Chapter 12. Using Sylow Theorems to Analyze Finite Groups*
    • 12.1. 𝑝-groups
    • 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 𝐹-isomorphisms
    • 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 𝑥ⁿ-𝑎
    • 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 (Odd-Numbered) Problems
    • Bibliography
    • Index
    • Back Cover
  • 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 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
  • Requests
     
     
    Review Copy – for reviewers who would like to review an AMS book
    Desk Copy – for instructors who have adopted an AMS textbook for a course
    Examination Copy – for faculty considering an AMS textbook for a course
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
Volume: 272017; 675 pp
MSC: Primary 00; Secondary 20; 13; 12;

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.

  • Cover
  • Title page
  • Contents
  • Preface
  • Part 1 . (Mostly Finite) Group Theory
  • Chapter 1. Four Basic Examples
  • 1.1. Symmetries of a Square
  • 1.2. 1-1 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 Inner-automorphisms^{⋆}
  • 11.7. More Problems and Projects
  • Chapter 12. Using Sylow Theorems to Analyze Finite Groups*
  • 12.1. 𝑝-groups
  • 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 𝐹-isomorphisms
  • 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 𝑥ⁿ-𝑎
  • 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 (Odd-Numbered) 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 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
Review Copy – for reviewers who would like to review an AMS book
Desk Copy – for instructors who have adopted an AMS textbook for a course
Examination Copy – for faculty considering an AMS textbook for a course
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
You may be interested in...
Please select which format for which you are requesting permissions.