Surely, $\mathbb{Z}_p$ and $\mathbb{Q}_p$ (and their extensions) are very important for algebra and number theory. Do they have any important applications outside of algebra (that I could easily explain to a student)? Here I do not demand the applications to be (purely) 'mathematical'; for example, I wonder whether padic numbers have applications to physics (outside of string theory?). Moreover, I am also interested in those applications that are partially 'algebraic', and yet important for some other parts of mathematics.

12$\begingroup$ There are applications of Witt vectors in algebraic topology. $\endgroup$– KConradDec 26, 2011 at 20:15

3$\begingroup$ @KConrad: It would make perfect sense to post this comment as an answer. $\endgroup$– Dmitri PavlovDec 27, 2011 at 8:03

$\begingroup$ @KConrad & Dmitri: It is related to my post below. Witt vectors show up because of their role in LubinTate Theory. $\endgroup$– Sean TilsonDec 27, 2011 at 23:46

$\begingroup$ Depends on what counts as "$p$adic". I like to think of two's complement arithmetic as truncated 2adics but you could also argue this point of view adds nothing to saying they are just arithmetic in $Z/2^n$. Similarly for computational applications of the 2adic Newton's method to solve equations. $\endgroup$– Dan PiponiJan 8, 2020 at 20:11
14 Answers
This won't qualify as something you can explain to undergraduate students, but nonarchimedean dynamics has recently seen a number of applications to classical complex dynamics. (Nonarchimedean is dynamics over a field with a nonarchimedean absolute value, but not specifically an extension of $\mathbb{Q}_p$.) I'll mention one beautiful example, which is a recent theorem of Matt Baker and Laura DeMarco. Let $$f_c(x) = x^2+c$$ be the usual quadratic polynomial, and for any starting value $a$, let $O_c(a)$ be the forward orbit of $a$ for the map $f_c$. That is, $$O_c(a) = \{a,f_c(a),f_c^2(a),f_c^3(a),...\}$$ where $f_c^n$ denotes the $n$'th iterate of $f_c$.
Theorem: Let $a$ and $b$ be complex numbers with $a^2\ne b^2$. Then $$\{c\in\mathbb{C} : O_c(a) \text{ and } O_c(b) \text{ are both finite}\}$$ is a finite set.
The proof is partly complex dynamics, partly equidistribution theorems (in both the complex and $p$adic settings), and partly a reduction step in which one works in Berkovich space over a nonarchimedean field. Note that the statement of the theorem is purely a statement about complex numbers, but the proof requires nontrivial methods from nonarchimedean analysis.
Monsky's theorem: it is not possible to dissect a square into an odd number of triangles of equal area.
Proof of this theorem is based on 2adic numbers.

2$\begingroup$ Are there any other applications in combinatorics? $\endgroup$– TurboOct 17, 2016 at 22:13
Skolem's theorem: Let $(u_n)_{n \in \mathbb N}$ be a sequence of complex numbers satisfying a linear relation. Then the set $Z=\{n \in \mathbb N,\ u_n=0\}$ is a finite union of arithmetic progressions in $\mathbb N$ (including arithmetic progression of ratio 0, that is points).
The only proof I know uses in an essential way $p$adic numbers (and elementary $p$adic analysis). You can explain the statement to an undergrad or even an highschool student, and the proof to anyone having the basic notions about $p$adic numbers.

2$\begingroup$ There is some more discussion on Terry Tao's blog here: terrytao.wordpress.com/2007/05/25/… $\endgroup$– j.c.Oct 29, 2013 at 15:15

3$\begingroup$ The late Alf van der Poorten and I published an expository article on Skolem's Theorem: Some problems concerning recurrence sequences, Amer Math Monthly 102 (October 1995) 698705, jstor.org/stable/2974639?seq=1#page_scan_tab_contents $\endgroup$ Oct 17, 2016 at 22:05
The (unsolved) HilbertSmith conjecture states that any locally compact group acting faithfully on a manifold has to be a Lie group: http://en.wikipedia.org/wiki/Hilbert%E2%80%93Smith_conjecture
However, it turns out that it is enough to prove this for $\mathbb{Z}_p$, and the conjecture follows proving that $\mathbb{Z}_p$ has no continuous faithful action on a manifold.

1$\begingroup$ Similarly to the reduction to $\mathbb{Z}_p$ is there a reduction to the manifold being say $\mathbb{R}$ or $S^1$? $\endgroup$ Jan 12, 2017 at 13:16
The $p$adics come up in homotopy theory. The main reason is because of their usefulness in the theory of formal group laws.
They are also relevant in certain parts of algebraic geometry, they are (one of) the first examples of completions.
References:
http://en.wikipedia.org/wiki/Formal_group#Lubin.E2.80.93Tate_formal_group_laws
http://arxiv.org/abs/1005.0119
http://arxiv.org/abs/0802.0996
The last one is supposed to tie in the others. Of course, this is all stuff that happened after Quillen's theorem and the work of many other people, such as Mike Hopkins, Jack Morava, Haynes Miller, Doug Ravenel, and Steve Wilson.

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11$\begingroup$ @Sean: I think Guntram was just invoking a witty variant of the "pics or it didn't happen" / "screenshot or it didn't happen" line that is prevalent elsewhere on the internet. In particular I think the answer to your question is "no" :) $\endgroup$ Dec 27, 2011 at 23:49

3$\begingroup$ Thanks for the explanation Kevin, I took it the way I would interpret the comment from anyone I didn't know. $\endgroup$ Dec 28, 2011 at 7:52
See the survey paper "On padic mathematical physics", by B. Dragovich, A. Yu. Khrennikov, S. V. Kozyrev and I. V. Volovich: Link
There is a journal devoted to padic numbers, padic analysis and applications: http://www.springer.com/mathematics/algebra/journal/12607
You can also google "spin glasses + padic numbers"

$\begingroup$ Yes, spin glasses are (is?) what caught my eye on the first page of the google results... $\endgroup$ Dec 26, 2011 at 20:12
The 2adic numbers have been used in cryptography, particularly in the analysis of feedback shift registers. See e.g.
 2adic shift registers, Klapper and Goresky, Fast Software Encryption 1993
 Cryptanalysis Based on 2Adic Rational Approximation, Klapper and Goresky, CRYPTO 1995
 Feedback shift registers, 2adic span, and combiners with memory, Feedback shift registers, 2adic span, and combiners with memory, Klapper and Goresky, J. Cryptology 1997
 Extended Games–Chan algorithm for the 2adic complexity of FCSRsequences, Meidl, Theoretical Computer Science 2003
 Feedback with carry shift registers synthesis with the Euclidean algorithm, Arnault, Berger, and Necer, IEEE Trans. Inf. Theory 2004
This article contains many references,
http://en.wikipedia.org/wiki/Padic_quantum_mechanics
though I don't know how many will count as "outside of string theory."
At a somewhat speculative level, you might look at the work of Andrei Khrennikov:
http://w3.msi.vxu.se/Personer/akhmasda/home.html
In particular, his home page mentions his work on:
 dynamical systems over $p$adic fields with applications for describing the process of thinking: "the system conscious subconscious as a padic dynamical processor."
 $p$adic dynamical systems for social sciences.

6$\begingroup$ right, see also > "transformative hermeneutics of padic gravity" $\endgroup$ Sep 5, 2019 at 2:26
In my answer to this related MO question, I tried to explain the utility of using $\mathbb{Q}_p$ based toy models in order to better understand questions in mathematical quantum field theory in physics.
Well, you can show that an old arithmetic coding algorithm can be reformulated in padic terms. For me this makes its integral (and the only working) variant easily understandable and makes it extendable for p > 2. See:
https://arxiv.org/abs/0704.0834
padic arithmetic coding in Contemporary Mathematics v. 508, 2010
I am not convinced by the applications of $p$adic numbers (or adèles) to theoretical physics, even though I am not a physicist. I think padic mathematical physics has so far nothing to do with real phenomena. But of course $p$adic analysis is useful in mathematics. $p$adic analysis has for instance natural applications in the $p$adic Langlands program. The basic idea of that program is to replace the field $\mathbb C$ by a $p$adic field when considering linear representations (of a $p$adic Lie group or a Galois group).
There are obvious applications of $p$adic numbers and adèles to analytic number theory via the (classical) Langlands program. These applications are not only algebraic, since they may for instance predict the analytic behaviour of $L$functions.
Another interesting example is the existence of a nice locally compact topology, defined by Berkovich, on $p$adic rigid manifolds (varieties over ${\mathbb C}_p$, the completion of the algebraic closure of ${\mathbb Q}_p$). You get in such a way varieties analogous to complex varieties. You can do very similar things like dynamics, dessins d'enfant, potential theory, integration of $1$forms, ... The following survey articles by Ducros (for french readers) are excellent:
Géométrie analytique $p$adique : la théorie de Berkovich, Gazette des Mathématiciens 111 (2007), 1227.
Espaces analytiques $p$adiques au sens de Berkovich, exposé 958 du Séminaire Bourbaki (mars 2006).
This is a very promising theory.

4$\begingroup$ Although I share your skepticism regarding part of the $p$adic physics literature, I don't think it is fair nor accurate to make such a blanket statement. See my answer to this MO question mathoverflow.net/questions/259155/padicnumbersinphysics for more on the usefulness of $p$adics in mathematical physics... $\endgroup$ Jan 12, 2017 at 12:23

6$\begingroup$ ...Some problems in math like making sense of quantum field theory are so difficult that one should try to understand as much as possible on simpler toy models where these problems are present but have a clear and clean formulation. This is what QFT models on $\mathbb{Q}_p$ provide. See the slides (in French) of my Colloquium at the University of Lyon for a pedagogical introduction to this circle of ideas. The link is: people.virginia.edu/~aa4cr/TalkLyonVfinal.pdf $\endgroup$ Jan 12, 2017 at 12:27

$\begingroup$ @AbdelmalekAbdesselam It seems that this link (as well as the link to your website in your profile) seem to redirect to some default page. Would you happen to have some working link to those slides available somewhere? $\endgroup$– WojowuSep 30, 2022 at 12:58

$\begingroup$ @Wojowu: My university stopped hosting personal pages, and I didn't have time to recreate my website on another platform. Some of it can be found at web.archive.org/web/20191228104704/http://people.virginia.edu/… However the Lyon slides you mentioned have not been captured by webachive. I can email them to you if you wish. $\endgroup$ Oct 2, 2022 at 17:31
Googling "applied padic analysis" returns obviously interesting results.

1$\begingroup$ But mostly motivated by number theory, right? $\endgroup$ Oct 17, 2016 at 16:46