# Roadmap to $p$-adic numbers: where a self-learner should look for references

TL;DR at the end of the question.

I’m currently trying to learn as much as possible about p-adic numbers. I’m not sure what is the most fascinating part of the theory, but the use of the adjective “p-adic” is quite wide-spread (I am aware of p-adic quantum mechanics, functional analysis or differential equations).

I would like to collect in one place materials that are worth reading and covering these numbers. It’s not surprising that in the internet it’s possible to find an introduction, but most of these documents don’t present everything and I am looking need research tools useful for a scientist. Unfortunately people at my university tend to be interested in analytical branches of maths, that’s why I’m posting here.

It seems that p-adic numbers remain quite unknown – even when building $\mathbb R$, the completion of $\mathbb Q$ with respect to usual absolute value, one rarely mentions that it’s possible to define another norm and get another field in the process, namely $\mathbb Q_p$.

In my opinion books are the best source of knowledge. I already skimmed through four texts:

1. p-adic numbers by F. Gouvea,
2. p-adic Numbers, Analysis and $\zeta$-Functions by N. Koblitz,
3. p-adic Analysis by A. Robert and last, but not least,
4. p-adic Analyisis and Mathematical Physics by Vladimirov, Volovich and Zelenov.

The problem is that they aren’t quite compatible with each other.

For example, Gouvea states that it’s hard to define an integral, Koblitz says that the p-adc integral should be the limit of Riemann sums: $$\int f\mu = \lim_N \sum_{a + (p^N) \subset X} f(x_{a,N}) \mu(a+(p^N))$$ as $a$ ranges from $0$ to $p^N – 1$ and $\mu$ is a p-adic measure – a distribution whose values are bounded by some constant for compact-open $U \subset X$). Robert admits that the Volkenborn integral will be more suitable, the last book defines normalized Haar measure that is invariant with respect to shifts and then develops the theory of integration in $\mathbb Q_p^n$. (I'm not sure if it's not similar to the point of view of Koblitz).

I’m also interested in reading articles (recently I have found Does it really converge? and it is pretty interesting) or learning the main theorems (by now I have discovered four: Ostrowski’s about absolute values on $Q$, Strassman’s about zeroes of power series, Hensel’s lemma and Hasse-Minkowski theorem about local-global principle - what are more?). I’m not familiar with advanced algebraic number theory (what are the adeles?!) but don’t struggle with topology or algebra at undergraduate level.

I would appreciate any recommendations for further hints how to gentle expand my knowledge. Final note: I'm aware of this question, but the answers aren't satisfying me: I don't speak French and dislike analysis.

TL;DR: I'm looking for materials (books, articles) about $p$-adic numbers at nonelementary level (possibly) suitable for an undergraduate, something more challenging than (usually recommended) book by Gouvea.

• Rational Quadratic Forms by Cassels. The most satisfying description on Hasse-Minkowski, along with how to calculate things. After that, if you learn how to use the Mass Formula for a genus of (positive) forms, and the representation counts for a genus representing numbers by Siegel's formula, you will be ahead of the game. I contiue to think that I would have made far more sense of Lie algebras if I had had quadratic forms first, perhaps in the guise of integral lattices. – Will Jagy May 25 '15 at 21:18
• You might look into “Local Fields and their Extensions” by Fesenko and Vostokov. – Lubin May 26 '15 at 2:58