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Introduction to the Theory of Numbers

Harold N. Shapiro
Dover Publications
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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Mehdi Hassani
, on

First of all, do not be misled by this book’s modest title. Indeed, this book is a fairly advanced version of what we might call “fundamental number theory.”

Usually, when we do serious research in number theory, we require some results from fundamental number theory, but soon we understand that almost all books with that title contain only some primal and formal topics. This books goes further.

Because the book is pitched at a higher level, as a whole it is not suitable for beginning undergraduates, but a good teacher will find a lot of good ideas and complementary comments. Essentially, the book is suitable for master-level students and first year graduates, especially those who are planning to work in number theory. It contains some very good and fresh exercises, sometimes including nice results. The author ends each chapter with remarkable historical notes concerning fundamental theorems and topics of number theory, and also with a list of related references, which mainly are classic. The expert author has his own fluent writing style.

The book has both algebraic and analytic flavor, with a slight preference for the analytic approach. For example, it includes elementary results on the distribution of prime numbers. It is written to be self-contained, so the author gives the proofs in detail. He tries to present topics and results in their mostly completed forms, and to do this, he doesn't hesitate to write long. As a restult, teaching through the whole book would take a long time. A sharp teacher might get around this by teaching the framework and then leaving details to the students.

I strongly recommend this book for students, teachers and researchers.

Mehdi Hassani is a co-tutelle Ph.D. student in Mathematics in the Institute for Advanced Studies in Basic Science in Zanjan, Iran, and the Université de Bordeaux I, under supervision of the professors M.M. Shahshahani and J-M. Deshouillers.


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