I was instantly drawn to the title of these volumes. I have for many years believed that a solid number theory course could serve the role currently played in many mathematics departments by a "Transition to higher mathematics" type of course, but with the advantage of unified content. The suggestion in the title that this was a text for a number theory course with that perspective was highly attractive.
That's not what this work is meant for.
On the other hand, what it is meant for is to present number theory as a unifying thread running through diverse areas of mathematics — and at this it succeeds. The author states at the outset that this is a work in theoretical rather than experimental or computational number theory, thus we find no treatment of the applications of number theory to cryptography. This is not, however, a deficiency. Part A covers all of the standard topics of a first course in number theory, but does so in far more detail than one normally sees. Many number theory books describe the development of number systems from the natural numbers up through the real or complex numbers — Coppel goes two steps further and reaches the octonions in chapter I, indicating that something ambitious is afoot here.
The end of Part A, and all of Part B , are where this book shines. We find here applications of number theory to such diverse fields as non-Euclidean geometry, dynamical systems, and linear algebra. The prime number theorem is presented here in full detail; so too are elliptic functions and ergodic theory as they illuminate number theory. These last-mentioned sections are, of course, intended for a more advanced reader, but the rewards for reading through to the end are immense.
Two things keep this set from easy use as a text: lack of numerical examples and of exercises — both gaps, of course, could be easily filled by a motivated instructor. However, for the right classroom of students, this would be a fine choice as a first number theory textbook. As a source for information on the "reach" of number theory into other areas of mathematics, it is an excellent work.
Mark Bollman (firstname.lastname@example.org) is an assistant professor of mathematics at Albion College in Michigan. His mathematical interests include number theory, probability, and geometry. His claim to be the only Project NExT fellow (Forest dot, 2002) who has taught both English composition and organic chemistry to college students has not, to his knowledge, been successfully contradicted.
Preface.- The Expanding Universe of Numbers.- Divisibility.- More on Divisibility.- Continued Fractions and their Uses.- Hadamard’s Determinant Problem.- Hensel’s P-Adic Numbers.- Notations.- Axioms.- Index.