*p*-adic Numbers: an Introduction

## by Fernando Q. Gouvêa

The
third edition of my introduction to the *p*-adic numbers was
published in July 2020. In addition to correcting all known typos, I
added a lot of new material: doing *p*-adic numbers with Sage and
GP, a section on visualizing the *p*-adic integers, a short
discussion of integration, etc. I hope I made it better!

This book is aimed at advanced undergraduates and graduate students
interested in beginning to learn about the *p*-adic numbers. It
is written in a way that I hope makes it usable for independent study
as well as for a class. The idea is to set you up for the deeper
treatments and applications you will find in graduate texts and
research monographs. An annotated list of some of what is available
appears in an appendix.

*p-adic Numbers* is part of the *Universitext* series from
Springer. The ISBNs are 978-3-030-47295-5 (ebook) and
978-3-030-47294-8 (softcover). You can
access
it at SpringerLink. It is also on Amazon, of course.

**Table of contents**

Introduction

On the Third Edition

**1. Apéritif:** Hensel’s Analogy, How to Compute,
Solving Congruences Modulo *p*^{n}, Other Examples.

**2. Foundations:** Absolute Values on a Field, Basic
Properties, Topology, Algebra.

**3. The ***p*-adic Numbers: Absolute Values on
**Q**, Completions.

**4. Exploring ****Q**_{p}: What We Already
Know, *p*-adic Integers, The Elements
of **Q**_{p}, What Does **Q**_{p}
Look Like, Hensel’s Lemma, Using Hensel’s Lemma,
Hensel’s Lemma for Polynomials, Local and Global.

**5. Elementary Analysis in ****Q**_{p}:
Sequences and Series, Functions, Continuity, Derivatives, Integrals,
Power Series, Functions Defined by Power Series, Strassman’s
Theorem, Logarithm and Exponential Functions, The Structure
of **Z**, The Binomial Series, Interpolation.

**6. Vector Spaces and Field Extensions:** Normed Vector Spaces
over Complete Valued Fields, Finite-dimensional Normed Vector Spaces,
Extending the *p*-adic Absolute Value, Finite Extensions of
**Q**_{p}, Classifying Extensions of
**Q**_{p}, Analysis, Example: Adjoining a
*p*-th Root of Unity, On to **C**_{p}.

**7. Analysis in ****C**_{p}: Almost
Everything Extends, Deeper Results on Polynomials and Power Series,
Entire Functions, Newton Polygons.

**8. Fun With Your New Head:** problems to explore.

**Appendix A. Sage and GP: A (Very) Quick Introduction **
**Appendix B. Hints, Solutions, and Comments on the Problems**

**Appendix C. A Brief Glance at the Literature **

I’ll try to collect here any typos or mistakes that are brought to my
attention. If you think you’ve found something,
send me an email!

I don’t have any errors yet for the third edition. There must be
some.