CONTENTS
Introduction = 1
1 Ap e ´ ritif = 5
1.1 Hensel's Analogy = 5
1.2 Solving Congruences Modulo pn = 12
1.3 Other Examples = 17
2 Foundations = 21
2.1 Absolute Values on a Field = 21
2.2 Basic Properties = 27
2.3 Topology = 29
2.4 Algebra = 37
3 p-adic Numbers = 41
3.1 Absolute Values on Q = 41
3.2 Completions = 47
3.3 Exploring Qp = 58
3.4 Hensel's Lemma = 67
3.5 Local and Global = 75
4 Elementary Analysis in Qp = 85
4.1 Sequences and Series = 86
4.2 Power Series = 88
4.3 Some Elementary Functions = 97
4.4 Interpolation = 109
5 Vector Spaces and Field Extensions = 117
5.1 Normed Vector Spaces over Complete Valued Field = 118
5.2 Finite-dimensional Normed Vector Spaces = 123
5.3 Finite Field Extensions = 127
5.4 Properties of Finite Extensions = 142
5.5 Analysis = 153
5.6 Example: Adjoining a p-th Root of Unity = 155
5.7 On to Cp = 161
6 Analysis in Cp = 171
6.1 Almost Everything Extends = 171
6.2 Derivatives = 175
6.3 Deeper Results on Polynomials and Power Series = 177
6.4 Entire Functions = 195
6.5 Newton Polygons = 199
6.6 Problems = 217
A. Hints and Comments on the Problems = 221
B. A Brief Glance at the Literature = 273
B.1 Texts = 273
B.2 Software = 274
B.3 Other books = 275
Bibliography = 277
Index = 279
