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

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## Fernando Q. Gouvêa: p-adic Numbers: an introduction

Published $\text {1997}$, Springer

ISBN 978-3-642-59058-0

### 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