Written by the leading Russian mathematician Igor Shafarevich, Discourses on Algebra is an advanced elementary algebra book intended to supplement the content of algebra courses in Russia, for students in grades 9-12 and their teachers. Its English translation can be used both for high school students and undergraduates interested in a deeper understanding of algebra topics, or in their preparation for mathematics competitions.
The material of the book is divided in seven chapters, covering three main topics: Numbers, Polynomials, and Sets. All the facts and proofs are very beautiful, some of them very well-known (such as "π is not Rational"), some of them probably new for most students (such as "Representation of Real Numbers as Decimal Fractions".)
All the sections in the book are followed by exercises (not easy!) intended to test both the understanding of the material presented and proved in the book, and the creativity and mathematical skills of the readers.
Several very important facts presented in the book have several proofs. As an example, Chapter 4 on Prime Numbers contains many proofs of the fact that "the Number of Prime Numbers is Infinite", including Euclid's (most beautiful?) proof and Euler's proof. The chapter ends with the non-trivial Chebyshev Inequality for the number of prime numbers not exceeding n, included as a supplement at the end of the chapter.
The last chapters, although presented as "algebra topics", are actually real analysis topics in disguise: Chapter 5 deals with Real Numbers and Polynomials (with sections such as "Limits and Infinite Sums" and "Real Roots of Polynomials") and Chapter 7 deals with Power Series. These topics fit very well with the rest of the content of the book and are so well explained that the students with no knowledge of real analysis (or even calculus) can understand and follow the proofs (and perhaps even solve the suggested problems).
What I found particularly attractive about this book are the historical notes, the references to many mathematicians and their work, as well as the 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 elementary) study of such topics as sets, polynomials, and numbers should read (pencil in hand!) this book. It may be particularly 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.
Mihaela Poplicher is assistant professor of mathematics at the University of Cincinnati. Her research interests include functional analysis, harmonic analysis, and complex analysis. She is also interested in the teaching of mathematics. Her email address is Mihaela.Poplicher@uc.edu.