p-adic numbers: an introduction

✍ Scribed by Fernando Q. Gouvêa

• Publisher: Springer
• Year: 2020
• Tongue: English
• Leaves: 371
• Series: Universitext
• Edition: 3
• Category: Library

✦ Synopsis

There are numbers of all kinds: rational, real, complex, p-adic, and more. The p-adic numbers are not as well known as the others, but they play a fundamental role in number theory and in other parts of mathematics, capturing information related to a chosen prime number p. They also allow us to use methods from calculus and analysis to obtain results in algebra and number theory.

This book is an elementary introduction to p-adic numbers. Most other books on the subject are written for more advanced students; this book provides an entryway to the subject for students with an undergraduate mathematics education. Readers who want to have an idea of and appreciation for the subject will probably find what they need in this book. Readers on the way to becoming experts can begin here before moving on to more advanced texts.

This third edition has been thoroughly revised to correct mistakes, make the exposition clearer, and call attention to significant aspects that are usually reserved for advanced books. The most important addition is the integration of mathematical software for computations with p-adic numbers and functions. A final chapter includes a selection of problems for further exploration.

✦ Table of Contents

Contents
Introduction
On the Third Edition
1 Apéritif
1.1 Hensel's Analogy
1.2 How to Compute
1.3 Solving Congruences Modulo pn
1.4 Other Examples
2 Foundations
2.1 Absolute Values on a Field
2.2 Basic Properties
2.3 Topology
2.4 Algebra
3 The p-adic Numbers
3.1 Absolute Values on Q
3.2 Completions
4 Exploring Qp
4.1 What We Already Know
4.2 p-adic Integers
4.3 The Elements of Qp
4.4 What Does Qp Look Like?
4.5 Hensel's Lemma
4.6 Using Hensel's Lemma
4.7 Hensel's Lemma for Polynomials
4.8 Local and Global
5 Elementary Analysis in Qp
5.1 Sequences and Series
5.2 Functions, Continuity, Derivatives
5.3 Integrals
5.4 Power Series
5.5 Functions Defined by Power Series
5.6 Strassman's Theorem
5.7 Logarithm and Exponential Functions
5.8 The Structure of Zp
5.9 The Binomial Series
5.10 Interpolation
6 Vector Spaces and Field Extensions
6.1 Normed Vector Spaces over Complete Valued Fields
6.2 Finite-dimensional Normed Vector Spaces
6.3 Extending the p-adic Absolute Value
6.4 Finite Extensions of Qp
6.5 Classifying Extensions of Qp
6.6 Analysis
6.7 Example: Adjoining a p-th Root of Unity
6.8 On to Cp
7 Analysis in Cp
7.1 Almost Everything Extends
7.2 Deeper Results on Polynomials and Power Series
7.3 Entire Functions
7.4 Newton Polygons
8 Fun With Your New Head
A Sage and GP: A (Very) Quick Introduction
A.1 Pari and GP
A.2 Sage
B Hints, Solutions, and Comments on the Problems
C A Brief Glance at the Literature
C.1 Textbooks
C.2 Other Books
Bibliography
Index

✦ Subjects

p-adic