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