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.

Preface

Some features of this book

A Note to Students

A Note to Instructors

Notation

1. Early Number Theory

2. Induction

3. Renaissance

4. Modular Arithmetic

5. Abstract Algebra

6. Arithmetic of Polynomials

7. Quotients, Fields, and Classical Problems

8. Cyclotomic Integers

9. Epilog References

Index

### About the Authors

**Al Cuoco** is Distinguished Scholar and Director of the Center for Mathematics Education at Education Development Center. He is lead author for The CME Project, a four-year NSF-funded high school curriculum, published by Pearson. He also co-directs Focus on Mathematics, a mathematics-science partnership that has established a mathematical community of mathematicians, teachers, and mathematics educators. The partnership evolved from his 25-year collaboration with Glenn Stevens (BU) on Boston University’s PROMYS for Teachers, a professional development program for teachers based on the Ross program (an immersion experience in mathematics). Al taught high school mathematics to a wide range of students in the Woburn, Massachusetts public schools from 1969 until 1993. A student of Ralph Greenberg, Cuoco holds a Ph.D. from Brandeis, with a thesis and research in Iwasawa theory. In addition to this book, MAA published his *Mathematical Connections: a Companion for Teachers and Others*. But his favorite publication is a 1991 paper in the *American Mathematical Monthly*, described by his wife as an attempt to explain a number system that no one understands with a picture that no one can see.

**Joseph Rotman** was born in Chicago on May 26, 1934. He studied at the University of Chicago, receiving the degrees BA, MA, and Ph.D. there in 1954, 1956, and 1959, respectively; his thesis director was Irving Kaplansky.

Rotman has been on the faculty of the mathematics department of the University of Illinois at Urbana-Champaign since 1959, with the following ranks: Research Associate 1959-1961; Assistant Professor 1961-1963; Associate Professor 1963-1968; Professor 1968-2004; Professor Emeritus 2004-present. He has held the following visiting appointments: Queen Mary College, London, England 1965, 1985; Aarhus University, Denmark, Summer 1970; Hebrew University, Jerusalem, Israel 1070; University of Padua, Italy, 1972; Technion, Israel Institute of Technology and Hebrew University, Jerusalem (Lady Davis Professor), 1977-78; Tel Aviv University, Israel, 1982; Bar Ilan University, Israel, Summer 1982; Annual visiting lecture, South African Mathematical Society, 1985; University of Oxford, England, 1990. Professor Rotman was an editor of *Proceedings of American Mathematical Society*, 1970, 1971; managing editor, 1972, 1973.

Aside from writing research articles, mostly in algebra, he has written the following textbooks: *Group Theory* 1965, 1973, 1984, 1995; *Homological Algebra* 1970, 1979, 2009; *Algebraic Topology* 1988; *Galois Theory* 1990, 1998; *Journey into Mathematics* 1998, 2007; *First Course in Abstract Algebr*a 1996, 2000, 2006; *Advanced Modern Algebra* 2002.