Timeline for Elementary results with p-adic numbers
Current License: CC BY-SA 3.0
28 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 17, 2023 at 3:54 | answer | added | Kapil | timeline score: 1 | |
| Jul 19, 2019 at 18:50 | answer | added | WhatsUp | timeline score: 5 | |
| Nov 23, 2011 at 19:16 | vote | accept | Daniele Turchetti | ||
| Nov 23, 2011 at 9:23 | comment | added | Daniele Turchetti | Actually I wanted to write "every point can be taken as center", which I think will reflect the fact that we can write explicitly B(c, r] for each center. Of course, in the comment it was way too heavy to write something like that (at least for people like me not used to write in a proper English). Thank you for pointing out that, anyway. About the Weil conjectures I definitively think you're right: it'll be too far for a single, or even double, talk. | |
| Nov 22, 2011 at 2:47 | comment | added | KConrad | Don't say every point in a ball is "its center" but rather is "a center". When one first learns about metric spaces one may get the implicit idea that the center and even the radius of a ball are unique, but it's not true in general and the audience will know from their own experience (when it's pointed out) that in the basic development of metric spaces one doesn't ever use such false facts anyway. | |
| Nov 22, 2011 at 2:38 | comment | added | KConrad | Daneile, it's hard to imagine your general audience of probabilists et al. could actually appreciate anything like the Weil conjectures since they probably do not comfortably understand what a finite field besides F_p really looks like in the first place. | |
| Nov 21, 2011 at 13:25 | comment | added | Daniele Turchetti | (It'll be funny to speak of Weil's conjectures too, but only if I manage to make it relevant to the context...) | |
| Nov 21, 2011 at 13:21 | history | edited | Daniele Turchetti | CC BY-SA 3.0 |
Retired the question asking if it's possible to find easy applications of p-adic analysis
|
| Nov 21, 2011 at 13:05 | comment | added | Daniele Turchetti | Thank you very much for your suggestions. I edited the question erasing the last question since it's clear that there are plenty of examples of what I'm looking for. I guess I'll change the title from "What are p-adic numbers?" to "Nice applications of p-adic analysis". I think I'll do the one about convergence to different rational numbers to shake their beliefs after having shown this odd geometry where every point of a ball is its center and the Diophantine problem to show that fuzzy and useful come together in this case. | |
| Nov 21, 2011 at 0:53 | answer | added | guy | timeline score: 0 | |
| Nov 20, 2011 at 23:33 | answer | added | user19414 | timeline score: 0 | |
| Nov 20, 2011 at 11:33 | answer | added | Chandan Singh Dalawat | timeline score: 13 | |
| Nov 20, 2011 at 8:29 | answer | added | Laurent Berger | timeline score: 5 | |
| Nov 20, 2011 at 6:16 | answer | added | Chandan Singh Dalawat | timeline score: 7 | |
| Nov 19, 2011 at 21:22 | answer | added | Aeryk | timeline score: 15 | |
| Nov 19, 2011 at 19:54 | comment | added | Laurent Berger | There's a theorem by Monsky (see jstor.org/stable/2317329) that a square cannot be divided into an odd number of triangles of equal areas. This is a purely geometrical statement, but the proof basically uses 2-adic numbers! | |
| Nov 19, 2011 at 19:51 | comment | added | Laurent Berger | Cassels' book "local fields" has a number of arithmetic applications of p-adic numbers. | |
| Nov 19, 2011 at 19:28 | comment | added | Phil Isett | @Chris Wuthrich -- I think that proving that $({\mathbb Z}/p^n {\mathbb Z})^\times$ is cyclic is really neat and does motivate the p-adic thinking. (One should note that for p=2 this fact fails.) I'm aware you can easily see that the image of the homomorphism $x \mapsto 1 + px + p^2 x^2/2! + \ldots$ (defined for p≥3), ${\mathbb Z} \to ({\mathbb Z}/p^n {\mathbb Z})^\times$ consists of all those elements congruent to 1 mod p. It's not quite using the p-adics yet. How do you use the logarithm? Is it simpler? | |
| Nov 19, 2011 at 18:58 | answer | added | Julien Puydt | timeline score: 3 | |
| Nov 19, 2011 at 18:43 | comment | added | Chris Wuthrich | One can prove that there is a primitive element modulo $p^n$, i.e. $(\mathbb{Z}/p^n\mathbb{Z})^{\times}$ is cyclic, for all $n\geq 1$ by using the case $n=1$ and the $p$-adic logarithm. Also, this explains that the discrete logarithm problem is not more difficult for $p^n$ than it is for $p$. | |
| Nov 19, 2011 at 17:22 | answer | added | Daniel Litt | timeline score: 20 | |
| Nov 19, 2011 at 17:16 | answer | added | KConrad | timeline score: 81 | |
| Nov 19, 2011 at 15:48 | comment | added | Maurizio Monge | Perhaps Hensel lemma can provide a quite elementary application. You can explaing to anybody what Newton's root-finding method is, and even display Newton's fractal created from the convergence data to a root over the complex numbers. So while p-adic numbers give you a system of numbers which is in some sense analogous to real numbers, the existence of a root can be ensured quite easily. | |
| Nov 19, 2011 at 15:48 | comment | added | Dan Piponi | If you have computer scientists in the audience, don't forget to mention the connection between two's complement arithmetic and the 2-adics. Practical application: using Newton-Raphson to compute multiplicative inverses modulo $2^n$, useful for fast exact integer division. | |
| Nov 19, 2011 at 14:28 | comment | added | Robert Garbary | One motivation for building the p-adic integers is to mimic the taylor series expansions for rational functions in the complex numbers. This could be a lecture by itself - start with some very elementary property of rational functions, and then write down the 'corresponding' property for the expansion of a rational number base p. | |
| Nov 19, 2011 at 14:22 | comment | added | Francesco Polizzi | Local-to-global results, for instance for quadratic and cubic forms (Hasse principle). See en.wikipedia.org/wiki/Hasse_principle and Chapter 1 of Serre's book "A course in arithmetic". | |
| Nov 19, 2011 at 13:56 | history | edited | Daniele Turchetti | CC BY-SA 3.0 |
edited body
|
| Nov 19, 2011 at 13:51 | history | asked | Daniele Turchetti | CC BY-SA 3.0 |