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Discovering Abstract Algebra
 
John K. Osoinach, Jr The University of Dallas, Irving, TX
Discovering Abstract Algebra
MAA Press: An Imprint of the American Mathematical Society
Softcover ISBN:  978-1-4704-6442-4
Product Code:  TEXT/67
List Price: $69.00
MAA Member Price: $51.75
AMS Member Price: $51.75
eBook ISBN:  978-1-4704-6734-0
Product Code:  TEXT/67.E
List Price: $69.00
MAA Member Price: $51.75
AMS Member Price: $51.75
Softcover ISBN:  978-1-4704-6442-4
eBook: ISBN:  978-1-4704-6734-0
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
Discovering Abstract Algebra
Click above image for expanded view
Discovering Abstract Algebra
John K. Osoinach, Jr The University of Dallas, Irving, TX
MAA Press: An Imprint of the American Mathematical Society
Softcover ISBN:  978-1-4704-6442-4
Product Code:  TEXT/67
List Price: $69.00
MAA Member Price: $51.75
AMS Member Price: $51.75
eBook ISBN:  978-1-4704-6734-0
Product Code:  TEXT/67.E
List Price: $69.00
MAA Member Price: $51.75
AMS Member Price: $51.75
Softcover ISBN:  978-1-4704-6442-4
eBook ISBN:  978-1-4704-6734-0
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 Details
     
     
    AMS/MAA Textbooks
    Volume: 672021; 199 pp
    MSC: Primary 00; Secondary 12; 13; 15; 20;

    Discovering Abstract Algebra takes an Inquiry-Based 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 two-semester course in abstract algebra, and will prepare students well for graduate-level study in algebra.

    Ancillaries:

    Readership

    Undergraduate 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 inquiry-based 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 student-driven experience that excels at giving motivated students the opportunity to explore and discover abstract algebra for themselves.

      Kevin Gerstle, Hillsdale College
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Desk Copy – for instructors who have adopted an AMS textbook for a course
    Instructor's Manual – for instructors who have adopted an AMS textbook for a course and need the instructor's manual
    Examination Copy – for faculty considering an AMS textbook for a course
    Accessibility – to request an alternate format of an AMS title
Volume: 672021; 199 pp
MSC: Primary 00; Secondary 12; 13; 15; 20;

Discovering Abstract Algebra takes an Inquiry-Based 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 two-semester course in abstract algebra, and will prepare students well for graduate-level study in algebra.

Ancillaries:

Readership

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 inquiry-based 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 student-driven experience that excels at giving motivated students the opportunity to explore and discover abstract algebra for themselves.

    Kevin Gerstle, Hillsdale College
Review Copy – for publishers of book reviews
Desk Copy – for instructors who have adopted an AMS textbook for a course
Instructor's Manual – for instructors who have adopted an AMS textbook for a course and need the instructor's manual
Examination Copy – for faculty considering an AMS textbook for a course
Accessibility – to request an alternate format of an AMS title
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