Learning Modern Algebra (Mathematical Association of America Textbooks) 1st Edition by Al Cuoco (PDF)

Description

Much of modern algebra arose from attempts to prove Fermat’s Last Theorem, which in turn has its roots in Diophantus’ classification of Pythagorean triples. This book, designed for prospective and practising mathematics teachers, makes explicit connections between the ideas of abstract algebra and the mathematics taught at high-school level. Algebraic concepts are presented in historical order, and the book also demonstrates how other important themes in algebra arose from questions related to teaching. The focus is on number theory, polynomials, and commutative rings. Group theory is introduced near the end of the text to explain why generalisations of the quadratic formula do not exist for polynomials of high degree, allowing the reader to appreciate the work of Galois and Abel. Results are motivated with specific examples, and applications range from the theory of repeating decimals to the use of imaginary quadratic fields to construct problems with rational solutions.

User’s Reviews

Editorial Reviews: Review 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. And not just high school algebra classes: there is, for example, a fairly lengthy and quite detailed section on straightedge and compass constructions, including statements and (at least partial, and often full) proofs of many sophisticated results regarding impossible constructions. This section is one of the ones in the book captioned “Connections”; every chapter of the book except for the last one contains a section with this title, the goal of which is to relate the material taught to what is taught in high school. These Connections sections not only cover the substantive mathematics that is actually taught but also sometimes address pedagogical issues as well, including the use of complex numbers and the norm function to create problems that, in the authors’ words, “come out nice,” or comments addressed to the teacher that help put the material covered in perspective. So, one potential course that this book could be used for is one with a title like Mathematics for High School Teachers, but unfortunately many colleges do not offer courses along these lines. …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, it is often conversational and humorous. Marginal notes appear frequently (as the authors point out, this seems particularly appropriate in a book of Fermat’s Last Theorem), and there are frequent remarks captioned “How to Think about it,” which offer informal advice to the student about ways of thinking of the material. There is also more attention paid than usual in books of this nature to historical discussion, and also — a feature I have noted in other books by Rotman — there is a lot of attention paid to etymology, the origin of mathematical terms. 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

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 ReviewsThis 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, CHOICEThis 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 thing about it, Historical notes, Etymology of notions and words. One finds chapters on: Early number theory (Greek, Euclid, Diophantus, trigonometry, integration), Induction, Renaissance (classical formulae, complex numbers, roots, powers, lattice point triangles), Modular arithmetic (codes, rings, patterns in decimal expansions), Abstract algebra, Arithmetic of polynomials, Quotients, fields, and classical problems (ruler-compass constructions) Cyclotomic integers (Gauss, Einstein, Fermat’s Last Theorem for exponent 3), Epilog references (Abel, Galois, solvability by radicals, groups, Wiles and Fermat’s last theorem, elliptic integrals, elliptic curves), and Appendices on basic linear algebra and a cyclotomic integer calculator. The above summing up, is by far not exhaustive. 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 Book Description A guide to modern algebra for mathematics teachers. It makes explicit connections between abstract algebra and high-school mathematics. Book Description Designed for prospective and practising mathematics teachers, this book makes explicit connections between the ideas of abstract algebra and the mathematics taught at high-school level. Algebraic concepts are presented in historical order, with focus on number theory, polynomials, and commutative rings and groups. About the Author Al Cuoco is Distinguished Scholar and Director of the Center for Mathematics Education at Education Development Center, Inc., (EDC). 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.Joseph Rotman has been on the faculty of the mathematics department of the University of Illinois, Urbana-Champaign since 1959, and has been Professor Emeritus since 2004. Professor Rotman was an editor of Proceedings of the American Mathematical Society (1970, 1971) and managing editor (1972, 1973). Aside from writing research articles, mostly on algebra, he has written textbooks on group theory, homological algebra, algebraic topology, Galois theory, and abstract algebra. Read more

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐Interesting Introduction to abstract algebra via FLT

⭐It has really good explanations and covers an ample selection of topics in abstract algebra.The historical notes are also quite helpful.

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