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

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

Published $\text {1997}$, Springer

ISBN 978-3-642-59058-0

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