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

Subject Matter

 * Analysis
 * $p$-adic Numbers

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