p-adic Numbers

1. Front Matter

   Pages i-vi

   PDF

2. Introduction

   • Fernando Q. Gouvêa

   Pages 1-5

3. Apéritif

   • Fernando Q. Gouvêa

   Pages 7-21

4. Foundations

   • Fernando Q. Gouvêa

   Pages 23-41

5. p-adic Numbers

   • Fernando Q. Gouvêa

   Pages 43-85

6. Elementary Analysis in ℚp

   • Fernando Q. Gouvêa

   Pages 87-132

7. Vector Spaces and Field Extensions

   • Fernando Q. Gouvêa

   Pages 133-185

8. Analysis in ℂp

   • Fernando Q. Gouvêa

   Pages 187-233

9. Back Matter

   Pages 235-305

   PDF

In the course of their undergraduate careers, most mathematics majors see little beyond "standard mathematics:" basic real and complex analysis, ab­ stract algebra, some differential geometry, etc. There are few adventures in other territories, and few opportunities to visit some of the more exotic cor­ ners of mathematics. The goal of this book is to offer such an opportunity, by way of a visit to the p-adic universe. Such a visit offers a glimpse of a part of mathematics which is both important and fun, and which also is something of a meeting point between algebra and analysis. Over the last century, p-adic numbers and p-adic analysis have come to playa central role in modern number theory. This importance comes from the fact that they afford a natural and powerful language for talking about congruences between integers, and allow the use of methods borrowed from calculus and analysis for studying such problems. More recently, p-adic num­ bers have shown up in other areas of mathematics, and even in physics.

• Algebra
• absolute values on fields
• calculus
• finite field
• number theory
• p-adic analysis
• p-adic numbers
• valuations

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)

• Department of Mathematics, Colby College, Waterville, USA

  Fernando Q. Gouvêa

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