Book:Fernando Q. Gouvea/p-adic Numbers: An Introduction

Subject Matter

 * Analysis
 * $p$-adic Numbers

Contents
Introduction

1$\quad$Apéritif
 * 1.1$\quad$Hensel’s Analogy
 * 1.2$\quad$Solving Congruences Modulo $p^n$
 * 1.3$\quad$Other Examples

2$\quad$Foundations
 * 2.1$\quad$Absolute Values on a Field
 * 2.2$\quad$Basic Properties
 * 2.3$\quad$Topology
 * 2.4$\quad$Algebra

3$\quad$$p$-adic Numbers
 * 3.1$\quad$Absolute Values on $\Q$
 * 3.2$\quad$Completions
 * 3.3$\quad$Exploring $\Q_p$
 * 3.4$\quad$Hensel’s Lemma
 * 3.5$\quad$Local and Global

4$\quad$Elementary Analysis in $\Q_p$
 * 4.1$\quad$Sequences and Series
 * 4.2$\quad$Functions, Continuity, Derivatives
 * 4.3$\quad$Power Series
 * 4.4$\quad$Functions Defined by Power Series
 * 4.5$\quad$Some Elementary Functions
 * 4.6$\quad$Interpolation

5$\quad$Vector Spaces and Field Extensions
 * 5.1$\quad$Normed Vector Spaces over Complete Valued Fields
 * 5.2$\quad$Finite-dimensional Normed Vector Spaces
 * 5.3$\quad$Finite Field Extensions
 * 5.4$\quad$Properties of Finite Extensions
 * 5.5$\quad$Analysis
 * 5.6$\quad$Example: Adjoining a $p$-th Root of Unity
 * 5.7$\quad$On to $\C_p$

6$\quad$Analysis in $\C_p$
 * 6.1$\quad$Almost Everything Extends
 * 6.2$\quad$Deeper Results on Polyomials and Power Series
 * 6.3$\quad$Entire Functions
 * 6.4$\quad$Newton Polygons
 * 6.5$\quad$Problems

A$\quad$Hints and Comments on the Problems

B$\quad$A Brief Glance at the Literature
 * B.1$\quad$Text
 * B.2$\quad$Software
 * B.3$\quad$Other Books

Bibliography

Index