# Book:Svetlana Katok/p-adic Analysis Compared with Real

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## Svetlana Katok:

## Svetlana Katok: *p-adic Analysis Compared with Real*

Published $\text {2007}$, **American Mathematical Society**

- ISBN 978-0821842201

### Subject Matter

### Contents

**Foreword: MASS and REU at Penn State University**

**Preface**

**Chapter 1. Arithmetic of the $p$-adic Numbers**

- 1.1. From $\Q$ or $\R$; the concept of completion
- Exercises 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 $\Q_p$
- Exercises 20-25

- 1.5. Arithmetic all operations in $\Q_p$
- Exercises 26-31

- 1.6. The $p$-adic expansion of rational numbers
- Exercises 32-34

- 1.7. Hensel’s Lemma and congruence
- 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 $\Q_g$ If $g$ is not a prime?
- Exercises 47-50

**Chapter 2. The Topology of $\Q_p$ 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 $\Z_p$
- Exercises 66-68

**Chapter 3. Elementary Analysis in $\Q_p$**

- 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. Isometrics of $\Q_p$
- Exercises 97-100

- 4.6. Interpolation
- Exercises 101-103

**Answers, Hints, and Solutions for Selected Exercises**

**Bibliography**

**Index**