Introductory book or other resource on $p$-adic numbers/number theory/analysis

I am having problems understanding $$p$$-adic numbers/$$p$$-adic number theory/$$p$$-adic analysis. I have tried some notes on the internet, but these notes were not helpful.

Can anyone suggest a book, lecture notes or video that gives a good introduction?

For example, I find Jeffrey Stopple (A Primer of Analytic Number Theory), John Stillwell (Elements of Number Theory) and Ian Stewart's (Galois Theory) books comprehensible, as these books are written in quite a descriptive way, in a friendly elaborate manner, so anything like these books will be good.

Postscript:

Is there any technical term for mathematical books which are descriptive and explain in an elementary way like the above-mentioned books of John Stillwell and Ian Stewart?

• Can you indicate what notes you already tried on the internet that were not helpful, so if someone recommends notes on the internet they won't be bringing up what you already looked at? – KCd Aug 6 '20 at 12:46

A few recommendations:

• Fernando Q. Gouvêa's $$p$$-adic numbers: An Introduction. This text gives a good introduction to the $$p$$-adic number system and the properties of the space of $$p$$-adic numbers, vector spaces over $$\mathbb{Q}_p$$, and the metrically completed algebraic closure of $$\mathbb{Q}_p$$. There is also some discussion of $$p$$-adic analysis near the end (e.g. power series and entire functions on $$\mathbb{C}_p$$).

I've used this book as a reference for a couple of undergraduate research projects on $$p$$-adic Brownian motion, and I think that the text is an appropriate introductory text for a general undergraduate audience. It would probably be best if they have already taken some real analysis, and familiarity with some basic complex analysis might help to motivate the material in the last chapter of the book, but nothing in the text really assumes that backgroud—a clever undergraduate with a solid calculus background and the ability to read and write proofs should be fine.

• Neal Koblitz's $$p$$-adic Numbers, $$p$$-adic Analysis, and Zeta-Functions is a little more advanced, and is more interested in the number theoretic aspects of $$p$$-adic analysis. Koblitz also introduces the reader to the $$p$$-adic numbers, but very quickly get's into some crunchier questions in algebraic geometry. There are some solid discussions of the Riemann zeta function and related problems—think "Tate's thesis, but with more exposition" (kind of).

I think that a student should have a relatively strong background in algebra before trying to tackle Koblitz. The constructions in the text are generally in terms of algebraic ideas: Koblitz works heavily with rings of polynomials with $$p$$-adic coefficients, and relies on intuition about the structure of such spaces. A background in algebraic geometry is not, strictly speaking, necessary, but knowing what a sheaf is wouldn't hurt (on the other hand, I suppose that one might take Koblitz's book as an introduction to these ideas, and use it to motivate students to learn algebraic geometry in greater detail). One might give this book to a clever undergraduate, but I think that it is probably more appropriate for a beginning graduate student.

• W. H. Schikof's *Ultrametric Calculus: An Introduction to $$p$$-adic Analysis is a solid introduction to $$p$$-adic analysis. It feels like it fills the same role in introducing the theory to students as Apostol's Mathematical Analysis or Spivak's Calculus—it gives a relatively gentle introduction to the material, but without holding the student's hand. The "meat" of the book consists of two chapters which give an overview of elementary calculus in $$\mathbb{Q}_p$$, but there is some discussion of ideas in functional analysis, too.

I get the feeling that this text was written with undergraduates in mind. I think that a student who has made it through the US-standard calculus curriculum could probably get started in a this book without too much difficulty, though if they have never seen a proof before, they will need help. A student with some background in analysis should have no difficulty working through most of Koblitz's text.

• I've used Koblitz's book for an undergraduate reading course on $p$-adic numbers, and it worked pretty well. – Daniel Hast Aug 12 '20 at 1:29
• @DanielHast That's good to know. Perhaps my assessment of the text is biased by my background as an analyst, rather than algebraist or number theorist. :) – Xander Henderson Aug 12 '20 at 1:44

There's a really nice book by Svetlana Katok called "p-adic analysis compared to Real."