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p-adic Numbers: An Introduction

Fernando Q. Gouvêa
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The Basic Library List Committee recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
Darren Glass
, on

Recently, I was asked by the MAA Basic Library List Committee to look through the books on The MAA’s Basic Library List, and to suggest any additions of books in Number Theory that might not already be on the list. To my eyes, the most glaring omission was a book by Fernando Gouvêa, originally published in 1993, entitled p-adic Numbers: An Introduction. Most recently, a third printing of the second edition of this book was made available in 2003 as part of Springer Verlag’s Universitext Series.

My own first exposure to the book came when I was an undergraduate and one of my friends came back from an REU and gave a talk on p-adic numbers. I was intrigued by one of the counterintuitive results that one gets when working in the p-adic topology — all triangles are isosceles, but none are equilateral! — and asked my friend where I could go to learn more. He told me that he had looked at several books and papers during his research, but that by far the best introduction was the book by Fernando Gouvêa. I checked it out of the library and found a book that was mathematically rich while still being clear and lively to read. In the intervening decade and a half, as I have spent much time reading about and working with p-adic numbers in my own research, I have read any number of treatments of the subject, and certainly several other sources come to mind that have various advantages in their different goals and perspectives. But if I had to recommend one book on the subject to a student — or even to a fully grown mathematician who had never played with p-adic numbers before — it would still be this book.

For those who don’t know, defining the p-adic numbers involves using an absolute value on rational numbers different from the standard absolute value and which in turn gives a different completion of the rational numbers than the ordinary real number system. More specifically, we define a new absolute value for any prime number p, by writing any rational number x in the form x=pn(a/b) where n is an integer and a and b are integers that are not divisible by p. After doing this, one defines the p-adic absolute value of x as |x|p= p–n. In other words, numbers that have many factors of p in the numerator are “small” and numbers with many factors of p in the denominator are “large.” The p-adic rational numbers are then the completion of the rational numbers with respect to this absolute value. (For people who prefer an algebraic approach, one can also define the p-adic integers as an inverse limit of the rings Z/pnZ and then define the p-adic rational numbers as the fraction field of this inverse limit).

It turns out that the p-adic rational numbers are similar to the real numbers in some senses (they are locally compact completions of the normal rational numbers, they are not algebraically closed) while being very different in other senses (the p-adic rationals are totally disconnected and are not ordered, for example). One of the main reasons that we care about p-adic numbers is that, by working in this new topology, one can use the analytic facts about the power series to learn about the algebraic properties of divisibility and answer number theoretic questions. This puts the subject right at the crossroads of algebra and analysis, although it also has its tentacles in number theory, topology, algebraic geometry, dynamical systems, and even theoretical physics, as some view it as a promising approach to studying the non-Archimedean geometry of space-time at small distances.

I would write more about the mathematics, but nothing I could write here would do as much justice to the subject as Gouvêa’s own writing. If you want to get a bit more of a taste of the subject, I would check out the 'Aperitif' of his book, most of which is (as of this writing) available to preview at Google Books

There are a number of strong books in Springer Verlag’s Universitext series, but (in my experience) many of them seem to aim a little too high or a little too low in their target audience. In my opinion, Gouvêa’s book hits the mark perfectly. The first few chapters have very little in the way of prerequisites, and the later chapters require only the basic topics from undergraduate courses in analysis and algebra. The book is filled with exercises, many of which have hints or comments in an appendix, and could easily be used as a textbook for a course or for an independent study.

As Gouvêa writes in his introduction, “our aim is sightseeing, rather than a scientific expedition” and to this end he includes an appendix discussing where the reader might go to learn more about the field, as well as some discussion of how one might use computers in their exploration (although in this latter regard, I should say that the book is starting to show its age: even in the last few years there has been much more done in this arena. See, for example, the work done by the folks at SAGE.) In this reviewer’s opinion, Gouvêa has succeeded admirably in taking a topic that is not standard in the undergraduate mathematics curriculum and writing a book accessible to undergraduates that allows its reader to play with some intriguing mathematics and explore a topic which is both fun and important.

Full Disclosure: Yes, the Fernando Gouvêa who wrote this book is the same Fernando Gouvêa who is the editor of MAA Reviews. However, the reader can rest assured that this reviewer would have said equally flattering things about the book even if it wasn’t written by his editor. Besides, I couldn’t think of anything that an editor could use to bribe his volunteer reviewers with (More prominent placing on the site? First crack at the new Keith Devlin?), so I didn’t even bother asking.

Darren Glass is an Associate Professor of Mathematics at Gettysburg College whose research interests include number theory, algebraic geometry, and cryptography. If you want to confirm that this review is real and under no influence by the editor, feel free to contact him at

Aperitif.- Foundations.- p-adic Numbers.- Elementary Analysis in Qp.- Vector Spaces and Field Extensions.- Analysis in Cp.- Hints and Comments on the Problems.- A Brief Glance at the Literature.