p-adic Numbers: An Introduction

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Springer Science & Business Media, Dec 6, 2012 - Mathematics - 306 pages
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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.
 

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Contents

Introduction
1
Foundations 23
22
padic Numbers
43
Elementary Analysis in Qp 87
86
Vector Spaces and Field Extensions
133
Analysis in Cp
187
A Hints and Comments on the Problems 235
234
B A Brief Glance at the Literature
291
Index 299
298
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