Discourses on Algebra 2003rd Edition by Igor R. Shafarevich (PDF)

Description

Using various examples this monograph shows that algebra is one of the most beautiful forms of mathematics. In doing so, it explains the basics of algebra, number theory, set theory and probability. The text presupposes very limited knowledge of mathematics, making it an ideal read for anybody new to the subject. The author, I.R. Shafarevich, is well-known across the world as one of the most outstanding mathematicians of this century as well as one of the most respected mathematical writers.

User’s Reviews

Editorial Reviews: Review From the reviews:”… What I found particularly attractive about this book are the historical notes, the references to many mathematicians and their work, as well as many original proofs included. In closing: I think that any student and any teacher interested in a deeper study of elementary (and maybe not so elementry) study of such topics as sets, polynomials, and numbers should read (pencil in hand!) this book. It may be particulary valuable for future teachers. The book is very well written, and it has detailed proofs and many exercises. Above all, this book will be remembered for its beauty and elegance.” (M. Poplicher, Read This! The MAA Online book review column (for complete review see http://www.maa.org/review/shafarevichdiscourses.html”… As the book progresses it becomes increasingly clear that the author has planned the development of ideas meticulously, not only so that he can constantly surprise the reader with the next turn of events, but also so that he can demonstrate unexpected links between results in algebra, number theory and probability. There are numerous ideas here for providing stimulating lessons for able pupils as well as for self-study, and the book would make a valuable addition to the school or department library.” (G. Leversha, The Mathematical Gazette, 88:511, 2004)”Since it happens so rarely, one must take notice when a great master of mathematics writes an elementary text. … Shafarevich takes the subject much farther than any high school text. … This book could form the basis for a great “transition to abstraction” course and should certainly have a role in programs training high school mathematics teachers. Summing up: Recommended. General readers; lower- and upper-division undergraduates; professionals.” (D.V. Feldman, Choice, June 2003)”… While written for students at the secondary level, the text goes quite thoroughly into foundational matters normally encountered at university … This book is highly recommended for teenagers with a strong desire to study mathematics and for secondary teachers seeking to contextualize what is on the school syllabus and to appreciate what some of their students will meeet in university mathematics.” (Edward J. Barbeau, Zentralblatt für Mathematik 1004.00001)”Discourses on Algebra is an advanced elementary algebra book intended to supplement the content of algebra courses … . All the sections in the book are followed by exercises intended to test both the understanding of the material presented and proved in the book, and the creativity and mathematical skills of the readers. … The book is very well written, and it has detailed proofs and many exercises. Above all, this book will be remembered for its beauty and elegance.” (Mihaela Poplicher, MAA Online, September, 2004)”In this introduction to algebra, the author aims to show that the subject is no less beautiful, elegant and logically coherent than Euclidean geometry. … it should appeal to able students and undergraduates who are looking for a more sustained and challenging course in elementary algebra. … There are numerous ideas here for providing stimulating lessons for able pupils as well as for self-study, and the book would make a valuable addition to the school or department library.” (Gerry Leversha, The Mathematical Gazette, 88:511), 2004)”In this book, the author – a famous Russian algebraist – argues with examples that algebra is no less beautiful, elegant, and logically cohesive than geometry. It contains an exposition of some rudiments of algebra, number theory, set theory and probability presupposing very limited knowledge of mathematics.” (Zentralblatt für Didaktik der Mathematik, June 2002)”In this book, ‘the elements of algebra as a field of contemporary mathematics are laid out, based on material bordering the school program as closely as possible’. The author would like algebra to appeal to students in the way that Euclidean geometry does. … the text goes quite thoroughly into foundational matters normally encountered at university. … This book is highly recommended for teenagers with a strong desire to study mathematics and for secondary teachers seeking to contextualize what is on the school syllabus and to appreciate what some their students will meet in university mathematics.” (Edward J. Barbeau, Zentralblatt MATH, 1004:4, 2003)

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

⭐Seldom does a master mathematician write a really elementary book. The only prerequisite for reading this one, written by one of Russia’s top 20th century mathematicians, is knowledge of high school mathematics and some familiarity with basic proof methods like reductio ad absurdum and mathematical induction. In reward, one gets the chance to learn a wide spectrum of different topics: basics of number theory, set theory and probability, together with some of the deeper results of relatively elementary algebra (like Sturm’s theorem). In fact, this book covers most of what is taught those days in “Discrete Mathematics” classes, except in this book those topics are presented in a unified and systematic manner, and not as a hodgepodge of different unrelated concepts. Many results that I have seen in other books are explained here much more deeply and at the same time more clearly. One example are the closed form formula for sums of different powers of consecutive integers from 1 to n. I have seen those derived in a lot of books for some special cases, but the steps in the derivation always looked a bit like tricks pulled out of nowhere. Not so here, the results are explained clearly, proved to the last detail and developed until they reach the most general form. This is the only place I know one can see this level of mathematical mastery without having almost any background in advanced mathematics. You only need pen and paper and the willingness to really work things out as you read. This would be the perfect book for an “Introduction to mathematics” or “Discrete mathematics” class, as soon as the student learns some basics of logic and proof, but it is really a delight for everyone enjoying mathematics. I am surprised this book isn’t more widely known, it is as good as “What is mathematics?”, the great classic by Richard Courant and Herbert Robbins.

⭐This is a brilliant text for ANYONE who is interested into delving deeper into the Spirit of Higher mathematics . I would recommend it to any high school student who wants to pursue Mathematics as his or her career . All said and done , the book is NOT an easy read , in the sense that you can sit back and relax and simply flip through the pages . If you want to learn Algebra , then , this is the book for you , provided you are willing to take a pen and paper and work out the problems as well as do the lemmas and proofs for yourself . The explanation is extremely lucid . Shaferavich was a brilliant teacher and belonged to the greats that the Soviet Era produced . It is a pity to see that the fabric of Mathematics Training is gradually being destroyed in the garb of reducing the syllabus . This book provides sufficient material to train the student to take on problems posed by the various entrance exams like ISI , CMI as well as RMO and INMO competitions .

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