**AMS/MAA Textbooks**

Volume: 23;
2013;
459 pp;
Hardcover

Print ISBN: 978-1-93951-201-7

Product Code: TEXT/23

List Price: $62.00

AMS Member Price: $46.50

MAA Member Price: $46.50

**Electronic ISBN: 978-1-61444-612-5
Product Code: TEXT/23.E**

List Price: $62.00

AMS Member Price: $46.50

MAA Member Price: $46.50

# Learning Modern Algebra: From Early Attempts to Prove Fermat’s Last Theorem

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*Al Cuoco; Joseph J. Rotman*

MAA Press: An Imprint of the American Mathematical Society

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.

An instructor's manual for this title is available
electronically. Please send email to textbooks@ams.org for more
information.

#### Reviews & Endorsements

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

# Table of Contents

## Learning Modern Algebra: From Early Attempts to Prove Fermat's Last Theorem

- front cover cover11
- copyright page ii3
- title page iii4
- Contents ix10
- Preface xiii14
- Notation xvii18
- Early Number Theory 122
- Induction 4566
- Renaissance 81102
- Modular Arithmetic 131152
- Abstract Algebra 191212
- Arithmetic of Polynomials 233254
- Quotients, Fields, and Classical Problems 277298
- Cyclotomic Integers 329350
- Epilog 379400
- Appendices 409430
- References 449470
- Index 451472
- About the Authors 459480