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Introduction 1 Aperitif 1.1 Hensers Analogy 1.2 Solving Congruences Modulo pn 1.3 Other Examples 2 Foundations 2.1 Absolute Values on a Field 2.2 Basic Properties 2.3 Topology 2.4 Algebra 3 p-adic Numbers 3.1 Absolute Values on Q 3.2 Completions 3.3 Exploring Qp 3.4 Hensel's Lemma 3.5 Local and Global 4 Elementary Analysis in Qp 4.1 Sequences and Series 4.2 Functions, Continuity, Derivatives 4.3 Power Series 4.4 Functions Defined by Power Series 4.5 Some Elementary Functions 4.6 Interpolation 5 Vector Spaces and Field Extensions 5.1 Normed Vector Spaces over Complete Valued Fields 5.2 Finite-dimensional Normed Vector Spaces 5.3 Finite Field Extensions 5.4 Properties of Finite Extensions 5.5 Analysis 5.6 Example: Adjoining a p-th Root of Unity 5.7 On to Cp 6 Analysis in Cp 6.1 Almost Everything Extends 6.2 Deeper Results on Polynomials and Power Series 6.3 Entire Functions 6.4 Newton Polygons 6.5 Problems A Hints and Comments on the Problems B A Brief Glance at the Literature B.1 Texts B.2 Software B.3 Other Books Bibliography Index