Softcover ISBN:  9781470464424 
Product Code:  TEXT/67 
List Price:  $69.00 
MAA Member Price:  $51.75 
AMS Member Price:  $51.75 
eBook ISBN:  9781470467340 
Product Code:  TEXT/67.E 
List Price:  $69.00 
MAA Member Price:  $51.75 
AMS Member Price:  $51.75 
Softcover ISBN:  9781470464424 
eBook: ISBN:  9781470467340 
Product Code:  TEXT/67.B 
List Price:  $138.00 $103.50 
MAA Member Price:  $103.50 $77.63 
AMS Member Price:  $103.50 $77.63 
Softcover ISBN:  9781470464424 
Product Code:  TEXT/67 
List Price:  $69.00 
MAA Member Price:  $51.75 
AMS Member Price:  $51.75 
eBook ISBN:  9781470467340 
Product Code:  TEXT/67.E 
List Price:  $69.00 
MAA Member Price:  $51.75 
AMS Member Price:  $51.75 
Softcover ISBN:  9781470464424 
eBook ISBN:  9781470467340 
Product Code:  TEXT/67.B 
List Price:  $138.00 $103.50 
MAA Member Price:  $103.50 $77.63 
AMS Member Price:  $103.50 $77.63 

Book DetailsAMS/MAA TextbooksVolume: 67; 2021; 199 ppMSC: Primary 00; Secondary 12; 13; 15; 20;
Discovering Abstract Algebra takes an InquiryBased Learning approach to the subject, leading students to discover for themselves its main themes and techniques. Concepts are introduced conversationally through extensive examples and student investigation before being formally defined. Students will develop skills in carefully making statements and writing proofs, while they simultaneously build a sense of ownership over the ideas and results. The book has been extensively tested and reinforced at points of common student misunderstanding or confusion, and includes a wealth of exercises at a variety of levels.
The contents were deliberately organized to follow the recommendations of the MAA's 2015 Curriculum Guide. The book is ideal for a one or twosemester course in abstract algebra, and will prepare students well for graduatelevel study in algebra.
Ancillaries:ReadershipUndergraduate students and researchers interested in an IBL approach to undergraduate algebra.

Table of Contents

Title page

Copyright

Contents

Acknowledgments

To the instructor

How to use this book

Class structure

Content and pace

Exercises

To the student

Part 1. Group theory

Chapter 1. Introduction

1.1. A brief backstory

1.2. Properties of the integers

Chapter 2. Binary operations

2.1. Closure

2.2. Binary tables

2.3. Isomorphic structures

Chapter 3. Groups

3.1. Basic properties of groups

3.2. Group notation

3.3. Group tables and the order of a group

Chapter 4. Subgroups and generating sets

4.1. Subgroups

4.2. The center of a group

4.3. Generating sets

Chapter 5. Applications of subgroups

5.1. Cosets

5.2. Lagrange’s theorem

5.3. Conjugation

Part 2. Types of groups

Chapter 6. Quotient groups

6.1. Homomorphisms and kernel

6.2. Normal subgroups

6.3. The natural projection homomorphism

Chapter 7. Cyclic groups

7.1. Properties of cyclic groups

7.2. Infinite cyclic groups

7.3. Finite cyclic groups

Chapter 8. Direct products

8.1. External direct products

8.2. Finitely generated abelian groups

Chapter 9. The isomorphism theorems

9.1. The first isomorphism theorem

9.2. Quotients of finitely generated abelian groups

9.3. The second and third isomorphism theorems

Chapter 10. The symmetric groups

10.1. Permutations

10.2. Dihedral groups

10.3. Cayley’s theorem

Chapter 11. Alternating groups

11.1. Orbits and cycles

11.2. Transpositions and the parity of a permutation

11.3. The alternating group

11.4. Generating sets for symmetric groups

11.5. The simplicity of 𝐴₅

Part 3. Ring theory

Chapter 12. Rings

12.1. Basic properties of rings

12.2. Homomorphisms

12.3. Polynomials

Chapter 13. Commutative rings

13.1. Integral domains

13.2. The Ring ℤ_{𝕟}

13.3. Polynomials over integral domains

Chapter 14. Fields

14.1. The field of quotients

14.2. The characteristic of a ring

14.3. Polynomials over a field

Chapter 15. Quotient rings

15.1. Ideals

15.2. Ideals in commutative rings

15.3. Ideals in polynomial rings

Part 4. Linear algebra

Chapter 16. Vector spaces

16.1. Basic properties of vector spaces

16.2. Linear combinations and span

16.3. Linear independence and bases

16.4. The dimension of a vector space

Chapter 17. Linear transformations

17.1. Bases and linear transformations

17.2. Rank and nullity

17.3. Eigenvectors

17.4. Linear operators

17.5. Dual spaces

Part 5. Field theory

Chapter 18. Extension fields

18.1. Degree of an extension

18.2. Simple extensions

18.3. Splitting fields

Chapter 19. Algebraic extensions

19.1. Algebraic elements

19.2. Number fields

19.3. Finite fields

Part 6. Intermediate group theory

Chapter 20. Group actions

20.1. Definitions and examples

20.2. Orbits and stabilizers

20.3. Counting orbits

Chapter 21. The Sylow theorems

21.1. The class equation

21.2. The normalizer

21.3. The Sylow theorems

21.4. Applications to simple groups

Appendices

Appendix A. Relations and functions

A.1. Equivalence relations

A.2. Functions

A.3. Bijections and inverse functions

Appendix B. Matrices

B.1. Matrix algebra

B.2. Matrix inverses

B.3. Determinants

Appendix C. Complex numbers

C.1. Complex arithmetic

C.2. The geometry of complex numbers

C.3. Complex solutions of equations

Index


Additional Material

Reviews

This book is clearly designed with inquirybased learning in mind, and in its introduction, suggests a variety of ways this book could be incorporated into more active classrooms settings through group work, presentations, and flipped classrooms. ...Discovering Abstract Algebra provides a more studentdriven experience that excels at giving motivated students the opportunity to explore and discover abstract algebra for themselves.
Kevin Gerstle, Hillsdale College


RequestsReview Copy – for publishers of book reviewsDesk Copy – for instructors who have adopted an AMS textbook for a courseInstructor's Manual – for instructors who have adopted an AMS textbook for a course and need the instructor's manualExamination Copy – for faculty considering an AMS textbook for a courseAccessibility – to request an alternate format of an AMS title
 Book Details
 Table of Contents
 Additional Material
 Reviews
 Requests
Discovering Abstract Algebra takes an InquiryBased Learning approach to the subject, leading students to discover for themselves its main themes and techniques. Concepts are introduced conversationally through extensive examples and student investigation before being formally defined. Students will develop skills in carefully making statements and writing proofs, while they simultaneously build a sense of ownership over the ideas and results. The book has been extensively tested and reinforced at points of common student misunderstanding or confusion, and includes a wealth of exercises at a variety of levels.
The contents were deliberately organized to follow the recommendations of the MAA's 2015 Curriculum Guide. The book is ideal for a one or twosemester course in abstract algebra, and will prepare students well for graduatelevel study in algebra.
Ancillaries:
Undergraduate students and researchers interested in an IBL approach to undergraduate algebra.

Title page

Copyright

Contents

Acknowledgments

To the instructor

How to use this book

Class structure

Content and pace

Exercises

To the student

Part 1. Group theory

Chapter 1. Introduction

1.1. A brief backstory

1.2. Properties of the integers

Chapter 2. Binary operations

2.1. Closure

2.2. Binary tables

2.3. Isomorphic structures

Chapter 3. Groups

3.1. Basic properties of groups

3.2. Group notation

3.3. Group tables and the order of a group

Chapter 4. Subgroups and generating sets

4.1. Subgroups

4.2. The center of a group

4.3. Generating sets

Chapter 5. Applications of subgroups

5.1. Cosets

5.2. Lagrange’s theorem

5.3. Conjugation

Part 2. Types of groups

Chapter 6. Quotient groups

6.1. Homomorphisms and kernel

6.2. Normal subgroups

6.3. The natural projection homomorphism

Chapter 7. Cyclic groups

7.1. Properties of cyclic groups

7.2. Infinite cyclic groups

7.3. Finite cyclic groups

Chapter 8. Direct products

8.1. External direct products

8.2. Finitely generated abelian groups

Chapter 9. The isomorphism theorems

9.1. The first isomorphism theorem

9.2. Quotients of finitely generated abelian groups

9.3. The second and third isomorphism theorems

Chapter 10. The symmetric groups

10.1. Permutations

10.2. Dihedral groups

10.3. Cayley’s theorem

Chapter 11. Alternating groups

11.1. Orbits and cycles

11.2. Transpositions and the parity of a permutation

11.3. The alternating group

11.4. Generating sets for symmetric groups

11.5. The simplicity of 𝐴₅

Part 3. Ring theory

Chapter 12. Rings

12.1. Basic properties of rings

12.2. Homomorphisms

12.3. Polynomials

Chapter 13. Commutative rings

13.1. Integral domains

13.2. The Ring ℤ_{𝕟}

13.3. Polynomials over integral domains

Chapter 14. Fields

14.1. The field of quotients

14.2. The characteristic of a ring

14.3. Polynomials over a field

Chapter 15. Quotient rings

15.1. Ideals

15.2. Ideals in commutative rings

15.3. Ideals in polynomial rings

Part 4. Linear algebra

Chapter 16. Vector spaces

16.1. Basic properties of vector spaces

16.2. Linear combinations and span

16.3. Linear independence and bases

16.4. The dimension of a vector space

Chapter 17. Linear transformations

17.1. Bases and linear transformations

17.2. Rank and nullity

17.3. Eigenvectors

17.4. Linear operators

17.5. Dual spaces

Part 5. Field theory

Chapter 18. Extension fields

18.1. Degree of an extension

18.2. Simple extensions

18.3. Splitting fields

Chapter 19. Algebraic extensions

19.1. Algebraic elements

19.2. Number fields

19.3. Finite fields

Part 6. Intermediate group theory

Chapter 20. Group actions

20.1. Definitions and examples

20.2. Orbits and stabilizers

20.3. Counting orbits

Chapter 21. The Sylow theorems

21.1. The class equation

21.2. The normalizer

21.3. The Sylow theorems

21.4. Applications to simple groups

Appendices

Appendix A. Relations and functions

A.1. Equivalence relations

A.2. Functions

A.3. Bijections and inverse functions

Appendix B. Matrices

B.1. Matrix algebra

B.2. Matrix inverses

B.3. Determinants

Appendix C. Complex numbers

C.1. Complex arithmetic

C.2. The geometry of complex numbers

C.3. Complex solutions of equations

Index

This book is clearly designed with inquirybased learning in mind, and in its introduction, suggests a variety of ways this book could be incorporated into more active classrooms settings through group work, presentations, and flipped classrooms. ...Discovering Abstract Algebra provides a more studentdriven experience that excels at giving motivated students the opportunity to explore and discover abstract algebra for themselves.
Kevin Gerstle, Hillsdale College