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.

The table of contents is reproduced below after the Errata. Note that the division of chapters is slightly different in this edition: the old chapter 3 was split into chapters 3 and 4, with the result that all the following chapters got a +1 change in numeration.

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!

1. Page 26, last paragraph: the reference should be to figure 1.2, not 1.3. There is no figure 1.3.
2. On page 29, we see 1 + 3 + 3² + ... + 3ⁿ + ... = ...33331, which is of course nonsense, since 3 is not even an allowable digit in Q₃. It should be ...111111.
3. Page 30, fourth displayed equation: this is supposed to be an infinite series, so it should end with ...
4. Page 30, Fact 1.4.1: As stated, this might suggest that the partial sums are integers, which of course they are not. So one should take “divisible by 2^M ” to mean that the numerator of the partial sum (in lowest terms) is divisible by 2^M.
5. Page 30, Problem 30: My hint for this is to look at D. P. Parent’s solution. Unfortunately, there are some problems with the solution there, so I have written my version of Parent’s proof. See also Problem 175.
6. Page 42, Problem 52: a right curly bracket is missing at the end of the ultrametric inequality.
7. Page 49, Proposition 2.4.1: | | is a “non-archimedean absolute value” and not, of course, a “non-archimedean valuation.”
8. Page 54, Proof of Lemma 3.1.2: there are several minor typos here. In the first paragraph, “a in B(a,e)” should be “x_n in B(a,e),” and “|x-a| < ε” should be “|x_n-a|< ε.” The second error occurs again in the next paragraph. Finally, in the sentence immediately after, “sequence” probably should be “sequences.”
9. Page 92 and preceding: I probably should have given Problem 120 as the main Hensel’s Lemma, since it is more useful than 4.5.2 or 4.5.3. I might also have added Hensel’s own version, which uses the discriminant of the polynomial to decide how close the approximate root must be.
10. Page 128, middle: the description of what happens in the classical case is incorrect. What I should have said is that in the classical case there is always a disk |x|<r on which the composed power series converges to the composed function. But we can’t just look at the value of g(x) (for example, we might have g(x)=0 for a value of x outside the radius of convergence of the composed series).
11. Page 138 and many others following after: good golly, the man’s name was Strassmann.
12. Page 229: The hint for problem 294 is not quite good enough. The problem is that there are two congruences to worry about: f(X) is congruent to g₁(X)h₁(X) (assumption (iii) in the theorem) and there are polynomials r(X) and s(X) such that g₁(X)r(X)+h₁(X)s(X) is congruent to 1 (assumption (ii) in the theorem). We need to choose δ so that both congruences are true modulo δ.
13. Pages 240–241. My statement of the p-adic Weierstrass preparation theorem is incorrect: condition (v) does not hold in general. What I should have said instead is that the norm of f(X)-g(X) is less than the norm of f(X), which implies that f(X) and g(X) have the same norm. This is clear from the proof, which is exactly the same as the proof of 7.2.3., so involves a sequence of polynomials g_i(X) that satisfy that condition for every i. See this Math.SE question, where Isaac Doosey-Shaw gives an explicit example of a series f(X) for which condition (v) cannot possibly hold. This means, alas, that the same correction needs to be made on pages 247, 268, and 269. Luckily, I don’t think I ever used the incorrect conclusion.
14. Page 306, solution for Problem 72: since we are assuming |x_1| < 1, its logarithm will be negative. So when we divide, the inequalities should reverse. Luckily, the argument still works..

Table of contents

Introduction

On the Third Edition

1. Apéritif: Hensel’s Analogy, How to Compute, Solving Congruences Modulo pⁿ, 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 B. Hints, Solutions, and Comments on the Problems

Appendix C. A Brief Glance at the Literature