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


Foreword: MASS and REU at Penn State University


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