$p$-adic Analysis Compared with Real ( Student Mathematical Library )

Publication series ：Student Mathematical Library

Author： Svetlana Katok  

Publisher： American Mathematical Society‎

Publication year： 2007

E-ISBN: 9781470421489

P-ISBN(Paperback): 9780821842201

Subject： O156.2 theory of algebraic numbers

Keyword： Number Theory

Language： ENG

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$p$-adic Analysis Compared with Real

Description

The book gives an introduction to $p$-adic numbers from the point of view of number theory, topology, and analysis. Compared to other books on the subject, its novelty is both a particularly balanced approach to these three points of view and an emphasis on topics accessible to undergraduates. In addition, several topics from real analysis and elementary topology which are not usually covered in undergraduate courses (totally disconnected spaces and Cantor sets, points of discontinuity of maps and the Baire Category Theorem, surjectivity of isometries of compact metric spaces) are also included in the book. They will enhance the reader's understanding of real analysis and intertwine the real and $p$-adic contexts of the book. The book is based on an advanced undergraduate course given by the author. The choice of the topic was motivated by the internal beauty of the subject of $p$-adic analysis, an unusual one in the undergraduate curriculum, and abundant opportunities to compare it with its much more familiar real counterpart. The book includes a large number of exercises. Answers, hints, and solutions for most of them appear at the end of the book. Well written, with obvious care for the reader, the book can be successfully used in a topic course or for self-study.

Chapter

Cover

Title

Copyright

Contents

Foreword: MASS and REU at Penn State University

Preface

Chapter 1. Arithmetic of the p-adic Number

§1.1. From Q to R; the concept of completion

Exercise 1–8

§1.2. Normed fields

Exercises 9–16

§1.3. Construction of the completion of a normed field

Exercises 17–19

§1.4. The field of p-adic numbers Q[sub(p)]

Exercises 20–25

§1.5. Arithmetical operations in Q[sub(p)]

Exercises 26–31

§1.6. The p-adic expansion of rational numbers

Exercises 32–34

§1.7. Hensel's Lemma and congruences

Exercises 35–44

§1.8. Algebraic properties of p-adic integers

§1.9. Metrics and norms on the rational numbers. Ostrowski's Theorem

Exercises 45–46

§1.10. A digression: what about Q[sub(g)] if g is not a prime?

Exercises 47–50

Chapter 2. The Topology of Q[sub(p)] vs. the Topology of R

§2.1. Elementary topological properties

Exercises 51–53

§2.2. Cantor sets

Exercises 54–65

§2.3. Euclidean models of Z[sub(p)]

Exercises 66–68

Chapter 3. Elementary Analysis in Q[sub(p)]

§3.1. Sequences and series

Exercises 69–73

§3.2. p-adic power series

Exercises 74–78

§3.3. Can a p-adic power series be analytically continued?

§3.4. Some elementary functions

Exercises 79–81

§3.5. Further properties of p-adic exponential and logarithm

§3.6. Zeros of p-adic power series

Exercises 82–83

Chapter 4. p-adic Functions

§4.1. Locally constant functions

Exercises 84–87

§4.2. Continuous and uniformly continuous functions

Exercises 88–90

§4.3. Points of discontinuity and the Baire Category Theorem

Exercises 91–96

§4.4. Differentiability of p-adic functions

§4.5. Isometries of Q[sub(p)]

Exercises 97–100

§4.6. Interpolation

Exercises 101–103

Answers, Hints, and Solutions for Selected Exercises

Bibliography

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Back Cover

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