p-adic Numbers: An Introduction (Universitext) (英語) ペーパーバック – 2013/10/4
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There are numbers of all kinds: rational, real, complex, p-adic. The p-adic numbers are less well known than the others, but they play a fundamental role in number theory and in other parts of mathematics. This elementary introduction offers a broad understanding of p-adic numbers.
From the reviews: "It is perhaps the most suitable text for beginners, and I shall definitely recommend it to anyone who asks me what a p-adic number is." --THE MATHEMATICAL GAZETTE
From the reviews:
"This is a well-written introduction to the world of p-adic numbers. The reader is led into the rich structure of the fields Qp and Cp in a beautiful balance between analytic and algebraic aspects. The overall conclusion is simple: an extraordinarily nice manner to introduce the uninitiated to the subject. Not only giving the background necessary to pursue the matter, but doing it in such a way that a healthy 'hands-on experience'is generated in the process." Mededelingen van het wiskundig genootschap
"It is perhaps the most suitable text for beginners, and I shall definitely recommend it to anyone who asks me what a p-adic number is." The Mathematical Gazette
From the reviews of the second edition:
“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. … 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.” (Darren Glass, The Mathematical Association of America, January, 2011)
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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.