p-adic Numbers: An Introduction (Universitext)

In particular, its proofs are rather clear, concise, and meaningful. There are only a couple points where Gouvea uses trickery (a la Rudin) to prove things, and he is honest enough to warn before they come. Indeed, the footnotes are quite entertaining, especially for one who has read enough math books to catch his jokes. Overall, the read is casual. It is good for independent reading.

The problems in the book are also worthy of praise. They are interspersed after proofs and in the middle of exposition, as ways to make sure the reader is following. Indeed, they are always pertinent and carefully planned. In addition, they point out ways in which mathematics authors often skip over details which the reader should actually verify for him/herself. Specifically, they perpare a student for reading graduate-level texts (which are notoriously full of this occurrence). Pedagogically, therefore, this is a very important tool, and useful book for a budding math student.

Finally, for the graduate student who wants to go down the path of p-adics, Gouvea does a good job of pointing the reader in the several different directions the literature can guide him/her. He gives references to other texts, giving just a taste of their contents, throughout. His eye towards further study is keen.

I must point out again that it is a joyful and entertaining read, in addition to being an exposition of a deeply fascinating (and deeply odd) area of math. This book, because of its clarity, its organization, its promotion of good mathematical reading skills, and its wonderful style occupies a spot on the very exclusive shelf of highest-quality texts in mathematics.


I only wish I could give more stars...   October 21, 2002
Cameron McLeman (Tucson, AZ United States)
18 out of 19 found this review helpful

This is the best introduction-level mathematics textbook I have ever read, and I have read (and own) many. Every theorem and definition is well-motivated, and problems for the reader are interspersed within the text to make sure that every subtle nuance of the exposition is understood. Though introductory, Gouvea manages to incorporate some relatively advanced topics, (as far as undergraduate mathematics goes) such as the Weierstrass Preparation Theorem, local rings, and analysis in C_p. All this and the topic at hand is fascinating to boot. p-adic numbers are perfect for anyone wishing to pick up an amazingly interesting topic not found in most undergraduate or graduate courses. In conclusion, I recommend this book to the set of people interested in p-adic numbers, and its complement.


enthousiasm-adique   July 11, 2000
Bahram Houchmandzadeh (France)
15 out of 18 found this review helpful

You're curious about beautifull topics in math, but reading a classical texbook, browsing through a list of lemma and therorems just get you sleep ? Read this one. The idea behind theorems, and why and how they came alive are so nicely exposed that this book can replace your favorite bed time one.