**Student Mathematical Library**

Volume: 37;
2007;
152 pp;
Softcover

MSC: Primary 11; 26; 12;

Print ISBN: 978-0-8218-4220-1

Product Code: STML/37

List Price: $32.00

Individual Price: $25.60

**Electronic ISBN: 978-1-4704-2148-9
Product Code: STML/37.E**

List Price: $32.00

Individual Price: $25.60

#### Supplemental Materials

# $p$-adic Analysis Compared with Real

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*Svetlana Katok*

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.

This book is published in cooperation with Mathematics Advanced Study Semesters

#### Table of Contents

# Table of Contents

## $p$-adic Analysis Compared with Real

- Cover Cover11 free
- Title iii4 free
- Copyright iv5 free
- Contents v6 free
- Foreword: MASS and REU at Penn State University ix10 free
- Preface xi12 free
- Chapter 1. Arithmetic of the p-adic Number 116 free
- §1.1. From Q to R; the concept of completion 217
- Exercise 1–8 520
- §1.2. Normed fields 621
- Exercises 9–16 1429
- §1.3. Construction of the completion of a normed field 1530
- Exercises 17–19 1934
- §1.4. The field of p-adic numbers Q[sub(p)] 1934
- Exercises 20–25 2641
- §1.5. Arithmetical operations in Q[sub(p)] 2742
- Exercises 26–31 3045
- §1.6. The p-adic expansion of rational numbers 3045
- Exercises 32–34 3348
- §1.7. Hensel's Lemma and congruences 3348
- Exercises 35–44 3853
- §1.8. Algebraic properties of p-adic integers 3954
- §1.9. Metrics and norms on the rational numbers. Ostrowski's Theorem 4358
- Exercises 45–46 4762
- §1.10. A digression: what about Q[sub(g)] if g is not a prime? 4762
- Exercises 47–50 5065

- Chapter 2. The Topology of Q[sub(p)] vs. the Topology of R 5368
- Chapter 3. Elementary Analysis in Q[sub(p)] 7590
- §3.1. Sequences and series 7590
- Exercises 69–73 8095
- §3.2. p-adic power series 8095
- Exercises 74–78 86101
- §3.3. Can a p-adic power series be analytically continued? 87102
- §3.4. Some elementary functions 89104
- Exercises 79–81 92107
- §3.5. Further properties of p-adic exponential and logarithm 92107
- §3.6. Zeros of p-adic power series 98113
- Exercises 82–83 102117

- Chapter 4. p-adic Functions 103118
- §4.1. Locally constant functions 103118
- Exercises 84–87 107122
- §4.2. Continuous and uniformly continuous functions 108123
- Exercises 88–90 112127
- §4.3. Points of discontinuity and the Baire Category Theorem 112127
- Exercises 91–96 115130
- §4.4. Differentiability of p-adic functions 116131
- §4.5. Isometries of Q[sub(p)] 121136
- Exercises 97–100 123138
- §4.6. Interpolation 123138
- Exercises 101–103 134149

- Answers, Hints, and Solutions for Selected Exercises 135150
- Bibliography 149164
- Index 151166
- Back Cover Back Cover1170

#### Readership

Undergraduate and graduate students interested in \(p\)-adic numbers.

#### Reviews

...the book gives a good impetus to students to study the "*p*-adic
worlds" more deeply. This role of the book is not only supported by carefully
selected material but also by the fact that it is written in a very lively and
lucid style.

-- European Mathematical Society Newsletter

I think that the reading of this book could animate some students to start to do research \(p\)-adic work. A good decision from my point of view!

-- Zentralblatt MATH