This book illustrates various elementary topics in number theory: irrationality and diophantine approximation, continued fractions and regular continued fractions, Euclidean domains, quadratic fields, primes and irreducibles, and Diophantine equations. The reader discovers the problem of representing numbers as sums of squares and learns about Padé approximants. There is a complete chapter devoted to different representations of real numbers.. Additionally, the book offers an introduction to exciting subjects in algebraic number theory. The last chapter focuses on different transcendence methods developed by Gelfond, Mahler, Schneider and Siegel.
The author combines rigor with clarity flawlessly, captivating the reader with the elegant simplicity of his prose. The methods used are deep and diverse. The book is both well written and pragmatic, while its material is pedagogical and comprehensive. There are clear explanations, concise definitions, and plenty of exercises along with their solutions.
There are twelve chapters, some of which are independent of each other. In every chapter one comes across a real gem. For example, the book discusses the famous problem of squaring the circle; the chapter contains the proof of Liouville’s famous theorem, talks about cyclotomic fields, and deduces the the Padé approximants of the exponential function by the means of confluent hypergeometric functions. There is even an explanation of how Padé approximants yield good Diophantine approximations and irrationality results.
Each chapter contains carefully and appropriately selected exercises that help the reader understand the theory better. They should not be skipped! Complete solutions can be found at the end of the book. Many of the solutions introduce/define additional concepts related to the topics of that particular chapter or are given along with examples and important remarks which complement them. The book contains exercises with the well-known: Möbius function, the Riemann zeta function, Dirichlet series, the Legendre symbol, etc. The reader will find them engaging, well-thought-out, and genuinely entertaining. In addition, the examples given throughout the book are both relevant and illuminating.
Originally written in French, the book has been translated into several languages, including in Japanese and English. The exhibition of the beauty of the number theory, achieved on the basis of its own simplicity and clarity, is one of the best features of this book. However, the reader should have previous knowledge of abstract algebra, calculus, and complex analysis. Thus, the book is well-suited for graduate and advanced undergraduate students. This is a fantastic and very useful book, although I would have not minded seeing a list of suggested further reading. I would highly recommend it to everyone who is seriously interested in number theory.
A native of Macedonia, Ana Momidic-Reyna has an M.S. in Mathematics and has also worked for the high energy physicists at Fermilab. While waiting for the opportunity to work on her Ph.D. in mathematics, she keeps up with the field by reading as many mathematics books as she can.