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Learning Modern Algebra: From Early Attempts to Prove Fermat’s Last Theorem
 
Learning Modern Algebra
MAA Press: An Imprint of the American Mathematical Society
Hardcover ISBN:  978-1-93951-201-7
Product Code:  TEXT/23
List Price: $75.00
MAA Member Price: $56.25
AMS Member Price: $56.25
eBook ISBN:  978-1-61444-612-5
Product Code:  TEXT/23.E
List Price: $69.00
MAA Member Price: $51.75
AMS Member Price: $51.75
Hardcover ISBN:  978-1-93951-201-7
eBook: ISBN:  978-1-61444-612-5
Product Code:  TEXT/23.B
List Price: $144.00 $109.50
MAA Member Price: $108.00 $82.13
AMS Member Price: $108.00 $82.13
Learning Modern Algebra
Click above image for expanded view
Learning Modern Algebra: From Early Attempts to Prove Fermat’s Last Theorem
MAA Press: An Imprint of the American Mathematical Society
Hardcover ISBN:  978-1-93951-201-7
Product Code:  TEXT/23
List Price: $75.00
MAA Member Price: $56.25
AMS Member Price: $56.25
eBook ISBN:  978-1-61444-612-5
Product Code:  TEXT/23.E
List Price: $69.00
MAA Member Price: $51.75
AMS Member Price: $51.75
Hardcover ISBN:  978-1-93951-201-7
eBook ISBN:  978-1-61444-612-5
Product Code:  TEXT/23.B
List Price: $144.00 $109.50
MAA Member Price: $108.00 $82.13
AMS Member Price: $108.00 $82.13
  • Book Details
     
     
    AMS/MAA Textbooks
    Volume: 232013; 459 pp

    Learning Modern Algebra aligns with the CBMS Mathematical Education of Teachers II recommendations, in both content and practice. It emphasizes rings and fields over groups, and it makes explicit connections between the ideas of abstract algebra and the mathematics used by high school teachers. It provides opportunities for prospective and practicing teachers to experience mathematics for themselves, before the formalities are developed, and it is explicit about the mathematical habits of mind that lie beneath the definitions and theorems.

    This book is designed for prospective and practicing high school mathematics teachers, but it can serve as a text for standard abstract algebra courses as well. The presentation is organized historically: the Babylonians introduced Pythagorean triples to teach the Pythagorean theorem; these were classified by Diophantus, and eventually this led Fermat to conjecture his Last Theorem. The text shows how much of modern algebra arose in attempts to prove this; it also shows how other important themes in algebra arose from questions related to teaching. Indeed, modern algebra is a very useful tool for teachers, with deep connections to the actual content of high school mathematics, as well as to the mathematics teachers use in their profession that doesn't necessarily “end up on the blackboard.”

    The focus is on number theory, polynomials, and commutative rings. Group theory is introduced near the end of the text to explain why generalizations of the quadratic formula do not exist for polynomials of high degree, allowing the reader to appreciate the more general work of Galois and Abel on roots of polynomials. Results and proofs are motivated with specific examples whenever possible, so that abstractions emerge from concrete experience. Applications range from the theory of repeating decimals to the use of imaginary quadratic fields to construct problems with rational solutions. While such applications are integrated throughout, each chapter also contains a section giving explicit connections between the content of the chapter and high school teaching.

    Ancillaries:

  • Table of Contents
     
     
    • Chapters
    • Chapter 1. Early Number Theory
    • Chapter 2. Induction
    • Chapter 3. Renaissance
    • Chapter 4. Modular Arithmetic
    • Chapter 5. Abstract Algebra
    • Chapter 6. Arithmetic of Polynomials
    • Chapter 7. Quotients, Fields, and Classical Problems
    • Chapter 8. Cyclotomic Integers
    • Epilog
    • A. Appendices
  • Additional Material
     
     
  • Reviews
     
     
    • This book covers abstract algebra from a historical perspective by using mathematics from attempts to prove Fermat's last theorem, as the title indicates. The target audience is high school mathematics teachers. However, typical undergraduate students will also derive great benefit by studying this text. The book is permeated with fascinating mathematical nuggets that are clearly explained

      D. P. Turner, CHOICE
    • This book is destined for college students in the U.S. who intend to teach mathematics in high school. The reviewer finds it even more apt as a text for algebra courses. Special features in the book are side notes given and printed prominently at the margins of the pages, like: How to think about it, Historical notes, Etymology of notions and words. ... The reviewer considers the book a refreshing read among the vast amount of books dealing with similar topics.

      Robert W. van der Waall, Zentrallblatt MATH
    • Although this book is designed for college students who want to teach in high school," its mathematical richness fits it admirably as a text for a first abstract algebra course or a handbook for assiduous students working on their own. While definitions, examples, theorems and their proofs are organized formally, the book is enhanced by substantial historical notes, advice on "how to think about it," marginal comments, connections and etymology that are designed to "balance experience and formality." The book is tightly organized with the goal of elucidating developments leading to the solution of the Fermat conjecture and the theory of solvability by radicals.

      E. J. Barbeau Mathematical Reviews
    • The primary intended audience of the book is future high school teachers. The authors take great pains to relate the material covered here to subjects that are taught in high school mathematics classes. ... In writing this book, the authors have obviously kept the needs of the student reader firmly in mind at all times. The writing style is not just clear; iit is often conversational and humorous. ... There are lots of exercises covering a wide range of difficulty, some with hints (but none with complete solutions) and there is a pretty good 39-entry bibliography. ... What might be a very interesting use for this book would be as a text for a senior seminar or “topics” course for students who already have some prior exposure to abstract algebra. And, of course, whatever may be the applicability of this book as a text for undergraduate course, it seems clear to me that it belongs in any good undergraduate library.

      Mark Hunacek MAA Reviews
  • 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: 232013; 459 pp

Learning Modern Algebra aligns with the CBMS Mathematical Education of Teachers II recommendations, in both content and practice. It emphasizes rings and fields over groups, and it makes explicit connections between the ideas of abstract algebra and the mathematics used by high school teachers. It provides opportunities for prospective and practicing teachers to experience mathematics for themselves, before the formalities are developed, and it is explicit about the mathematical habits of mind that lie beneath the definitions and theorems.

This book is designed for prospective and practicing high school mathematics teachers, but it can serve as a text for standard abstract algebra courses as well. The presentation is organized historically: the Babylonians introduced Pythagorean triples to teach the Pythagorean theorem; these were classified by Diophantus, and eventually this led Fermat to conjecture his Last Theorem. The text shows how much of modern algebra arose in attempts to prove this; it also shows how other important themes in algebra arose from questions related to teaching. Indeed, modern algebra is a very useful tool for teachers, with deep connections to the actual content of high school mathematics, as well as to the mathematics teachers use in their profession that doesn't necessarily “end up on the blackboard.”

The focus is on number theory, polynomials, and commutative rings. Group theory is introduced near the end of the text to explain why generalizations of the quadratic formula do not exist for polynomials of high degree, allowing the reader to appreciate the more general work of Galois and Abel on roots of polynomials. Results and proofs are motivated with specific examples whenever possible, so that abstractions emerge from concrete experience. Applications range from the theory of repeating decimals to the use of imaginary quadratic fields to construct problems with rational solutions. While such applications are integrated throughout, each chapter also contains a section giving explicit connections between the content of the chapter and high school teaching.

Ancillaries:

  • Chapters
  • Chapter 1. Early Number Theory
  • Chapter 2. Induction
  • Chapter 3. Renaissance
  • Chapter 4. Modular Arithmetic
  • Chapter 5. Abstract Algebra
  • Chapter 6. Arithmetic of Polynomials
  • Chapter 7. Quotients, Fields, and Classical Problems
  • Chapter 8. Cyclotomic Integers
  • Epilog
  • A. Appendices
  • This book covers abstract algebra from a historical perspective by using mathematics from attempts to prove Fermat's last theorem, as the title indicates. The target audience is high school mathematics teachers. However, typical undergraduate students will also derive great benefit by studying this text. The book is permeated with fascinating mathematical nuggets that are clearly explained

    D. P. Turner, CHOICE
  • This book is destined for college students in the U.S. who intend to teach mathematics in high school. The reviewer finds it even more apt as a text for algebra courses. Special features in the book are side notes given and printed prominently at the margins of the pages, like: How to think about it, Historical notes, Etymology of notions and words. ... The reviewer considers the book a refreshing read among the vast amount of books dealing with similar topics.

    Robert W. van der Waall, Zentrallblatt MATH
  • Although this book is designed for college students who want to teach in high school," its mathematical richness fits it admirably as a text for a first abstract algebra course or a handbook for assiduous students working on their own. While definitions, examples, theorems and their proofs are organized formally, the book is enhanced by substantial historical notes, advice on "how to think about it," marginal comments, connections and etymology that are designed to "balance experience and formality." The book is tightly organized with the goal of elucidating developments leading to the solution of the Fermat conjecture and the theory of solvability by radicals.

    E. J. Barbeau Mathematical Reviews
  • The primary intended audience of the book is future high school teachers. The authors take great pains to relate the material covered here to subjects that are taught in high school mathematics classes. ... In writing this book, the authors have obviously kept the needs of the student reader firmly in mind at all times. The writing style is not just clear; iit is often conversational and humorous. ... There are lots of exercises covering a wide range of difficulty, some with hints (but none with complete solutions) and there is a pretty good 39-entry bibliography. ... What might be a very interesting use for this book would be as a text for a senior seminar or “topics” course for students who already have some prior exposure to abstract algebra. And, of course, whatever may be the applicability of this book as a text for undergraduate course, it seems clear to me that it belongs in any good undergraduate library.

    Mark Hunacek MAA Reviews
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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