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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 p-adic 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.

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There are applications of Witt vectors in algebraic topology. –  KConrad Dec 26 '11 at 20:15
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@KConrad: It would make perfect sense to post this comment as an answer. –  Dmitri Pavlov Dec 27 '11 at 8:03
    
@KConrad & Dmitri: It is related to my post below. Witt vectors show up because of their role in Lubin-Tate Theory. –  Sean Tilson Dec 27 '11 at 23:46
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11 Answers

This won't qualify as something you can explain to undergraduate students, but non-archimedean dynamics has recently seen a number of applications to classical complex dynamics. (Non-archimedean is dynamics over a field with a non-archimedean 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 non-archimedean field. Note that the statement of the theorem is purely a statement about complex numbers, but the proof requires non-trivial methods from non-archimedean analysis.

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The (unsolved) Hilbert-Smith 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.

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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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References or it didn't happen. –  Guntram Dec 27 '11 at 8:52
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are you serious? –  Sean Tilson Dec 27 '11 at 17:53
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@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" :-) –  Kevin Buzzard Dec 27 '11 at 23:49
    
Thanks for the explanation Kevin, I took it the way I would interpret the comment from anyone I didn't know. –  Sean Tilson Dec 28 '11 at 7:52
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See the survey paper "On p-adic mathematical physics", by B. Dragovich, A. Yu. Khrennikov, S. V. Kozyrev and I. V. Volovich: http://www.springerlink.com/content/j16m84014r878034/

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This article contains many references,

http://en.wikipedia.org/wiki/P-adic_quantum_mechanics

though I don't know how many will count as "outside of string theory."

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There is a journal devoted to p-adic numbers, p-adic analysis and applications: http://www.springer.com/mathematics/algebra/journal/12607

You can also google "spin glasses + p-adic numbers"

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Yes, spin glasses are (is?) what caught my eye on the first page of the google results... –  Igor Rivin Dec 26 '11 at 20:12
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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 2-adic numbers.

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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 high-school student, and the proof to anyone having the basic notions about $p$-adic numbers.

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There is some more discussion on Terry Tao's blog here: terrytao.wordpress.com/2007/05/25/… –  j.c. Oct 29 '13 at 15:15
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Googling "applied p-adic analysis" returns obviously interesting results.

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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 p-adic dynamical processor."
  • $p$-adic dynamical systems for social sciences.
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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 p-adic 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), 12-27.

Espaces analytiques $p$-adiques au sens de Berkovich, exposé 958 du Séminaire Bourbaki (mars 2006).

This is a very promissing theory.

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