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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.

Svetlana Katok: Pennsylvania State University, University Park, PA

Cover 1
Title 4
Copyright 5
Contents 6
Foreword: MASS and REU at Penn State University 10
Preface 12
Chapter 1. Arithmetic of the p-adic Number 16
§1.1. From Q to R; the concept of completion 17
Exercise 1–8 20
§1.2. Normed fields 21
Exercises 9–16 29
§1.3. Construction of the completion of a normed field 30
Exercises 17–19 34
§1.4. The field of p-adic numbers Q[sub(p)] 34
Exercises 20–25 41
§1.5. Arithmetical operations in Q[sub(p)] 42
Exercises 26–31 45
§1.6. The p-adic expansion of rational numbers 45
Exercises 32–34 48
§1.7. Hensel's Lemma and congruences 48
Exercises 35–44 53
§1.8. Algebraic properties of p-adic integers 54
§1.9. Metrics and norms on the rational numbers. Ostrowski's Theorem 58
Exercises 45–46 62
§1.10. A digression: what about Q[sub(g)] if g is not a prime? 62
Exercises 47–50 65
Chapter 2. The Topology of Q[sub(p)] vs. the Topology of R 68
§2.1. Elementary topological properties 68
Exercises 51–53 75
§2.2. Cantor sets 75
Exercises 54–65 83
§2.3. Euclidean models of Z[sub(p)] 84
Exercises 66–68 88
Chapter 3. Elementary Analysis in Q[sub(p)] 90
§3.1. Sequences and series 90
Exercises 69–73 95
§3.2. p-adic power series 95
Exercises 74–78 101
§3.3. Can a p-adic power series be analytically continued? 102
§3.4. Some elementary functions 104
Exercises 79–81 107
§3.5. Further properties of p-adic exponential and logarithm 107
§3.6. Zeros of p-adic power series 113
Exercises 82–83 117
Chapter 4. p-adic Functions 118
§4.1. Locally constant functions 118
Exercises 84–87 122
§4.2. Continuous and uniformly continuous functions 123
Exercises 88–90 127
§4.3. Points of discontinuity and the Baire Category Theorem 127
Exercises 91–96 130
§4.4. Differentiability of p-adic functions 131
§4.5. Isometries of Q[sub(p)] 136
Exercises 97–100 138
§4.6. Interpolation 138
Exercises 101–103 149
Answers, Hints, and Solutions for Selected Exercises 150
Bibliography 164
Index 166
Back Cover 170
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