Timeline for How to picture $\mathbb{C}_p$?
Current License: CC BY-SA 4.0
27 events
| when toggle format | what | by | license | comment | |
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| Jan 9 at 12:30 | comment | added | Weier | "this picture continued infinitely "upward," Said like this, it looks very symmetric, as if we should allow an infinite numbers of decimals in the p-adic extension of a number in $\mathbb{Q}_p$. | |
| Jul 28, 2018 at 18:52 | history | edited | Daniel Litt | CC BY-SA 4.0 |
fixed pictures
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| Mar 10, 2017 at 9:42 | history | edited | CommunityBot |
replaced http://wwwmath.uni-muenster.de/ with https://wwwmath.uni-muenster.de/
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| Mar 10, 2017 at 9:42 | history | edited | CommunityBot |
replaced http://math.stanford.edu/ with https://math.stanford.edu/
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| Jun 7, 2012 at 18:58 | comment | added | Daniel Litt | Having looked at your other questions, however, I would guess you already know this :). | |
| Jun 7, 2012 at 18:56 | comment | added | Daniel Litt | @temp: Well, the Galois theory of F_p is very simple...as for visualizing F_q, I'm not really sure I know how to do that, since there's no topology. But I can promise that extensions of F_p do not feel so mysterious after a while (compared to e.g. extensions of Q or Q_p). If I were to venture a guess as to the psychological reason for this, it would be that understanding Abelian extensions is much easier than understanding general extensions, and all extensions of F_p are Abelian. | |
| Jun 7, 2012 at 7:08 | comment | added | temp | I can see the picture for Z_p and Q_p, but for unramified extensions, I feel like the reason the picture get messy for me is that I don't have a good picture for extension of finite fields! Do you have a way to visualize F_q? | |
| Jun 7, 2012 at 6:45 | comment | added | temp | I'm really curious, where does this visualization first come from? | |
| Jan 14, 2011 at 16:56 | comment | added | Daniel Litt | In particular, keep in mind, the branches are not extensions but are instead representatives of elements of the quotient by powers of the maximal ideal; I'm not sure one can see the difference between tame and wild ramification in this picture. | |
| Jan 14, 2011 at 16:49 | comment | added | Daniel Litt | You're right--I don't quite now how to represent this though. | |
| Jan 14, 2011 at 13:05 | comment | added | Maurizio Monge | Hello, while i like a lot you pictures, and say that they match more or less to the idea i always had about p-adic number and their extensions, i'm not sure about one thing. In particular the branches at points of height $a/bp^k$ for $k>0$ and $(ab,p)=1$ should correspond to totally ramified extensions, and usually there is a lot of them, not just one possibility obtained taking a root of the uniformizer. On the other hand for what concern the tamely ramified extensions, they are unique after taking the compositum with the maximal unramified extension, and the representation works perfectly. | |
| Jan 14, 2011 at 6:04 | comment | added | Alex B. | While the picture really is pretty funky, I am totally amazed by the thought that it might help somebody do arithmetic in $\mathbb{C}_p$ or even in $\mathbb{Z}_p$. Still +1 for the effort. | |
| Jan 14, 2011 at 6:02 | comment | added | Pete L. Clark | Wow. I actually gave this answer a +1 for the first time after seeing the second picture. As I wrote in my answer, I am in general not a very visual or pictorial thinker (total number of pictures in all of my papers so far: $0$), but...gadzooks, that's a cool graphic. And the tree structure is made especially vivid here, so (even!) I see actual mathematical content. | |
| Jan 14, 2011 at 5:10 | comment | added | Phillip Williams | Wow I like your new picture even more...I appreciate the effort in making it too, and the explanation. | |
| Jan 14, 2011 at 4:15 | history | edited | Daniel Litt | CC BY-SA 2.5 |
Added another picture and more remarks on extensions.
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| Jan 13, 2011 at 21:03 | comment | added | Daniel Litt | @Georges: I really appreciate your remark, though I do not yet have students :). | |
| Jan 13, 2011 at 16:14 | comment | added | Phillip Williams | I like this picture; thanks! I'll have to think about it more... | |
| Jan 13, 2011 at 8:52 | comment | added | Georges Elencwajg | Dear Daniel, thank you for the wonderful explanations and picture: I had never seen anything like that. American students, yours and Pete's for example, are very lucky to have teachers who explain mathematics in such a vividly visual way. | |
| Jan 13, 2011 at 5:50 | comment | added | Daniel Litt | (Just in case that wasn't clear, that last comment was in response to the first of your last two.) And of course it's good to view these pictures with some skepticism--I like the term "corona" though. | |
| Jan 13, 2011 at 5:48 | comment | added | Daniel Litt | That's absolutely true, and unfortunate--I think Bill Thurston laments this in one of his essays. That said, I at least found the picture of $\mathbb{Z}_p$ useful, and I particularly like the descriptions of ramified and unramified extensions (which I've surprisingly never seen described or drawn anywhere). So hopefully someone else will find this useful, and if not, no harm done I hope. | |
| Jan 13, 2011 at 5:46 | comment | added | Pete L. Clark | And, in a totally different direction: when I draw the picture of $p$-adic disks for students, I actually put the "blue disk" (i.e., the one corresponding to the maximal ideal) in the middle and draw the others around it. I point to the blue one and say "this is the open unit disk", then I point to the union of the others and say "this is the corona". Then I explain that when the field is locally compact, a positive proportion (tending to infinity with $p$) of the measure lies on the corona. And then I make sure to make fun of the picture a bit so that no one takes it too seriously. | |
| Jan 13, 2011 at 5:42 | comment | added | Pete L. Clark | @Daniel: if you'll permit me to say so -- along the lines of the answer I've since added, I think the issue is at least partly that when we try to describe our hard-earned intuition to others, it often comes out in distorted and less than helpful ways. | |
| Jan 13, 2011 at 5:39 | history | edited | Daniel Litt | CC BY-SA 2.5 |
Stressed an important point.
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| Jan 13, 2011 at 5:34 | comment | added | Daniel Litt | @Pete L. Clark: I've edited in agreement with your remark. The notion of "depth" I had in mind was given by the valuation, and I think was influenced by the picture I have in mind of $\operatorname{Spec} \mathbb{Z}_p$ as having a tower of closed subschemes corresponding to the powers of the maximal ideal. But I agree that this view is not particularly helpful. | |
| Jan 13, 2011 at 5:32 | history | edited | Daniel Litt | CC BY-SA 2.5 |
Edited as per Pete Clark's comments.
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| Jan 13, 2011 at 5:13 | comment | added | Pete L. Clark | "Now zero has the $p$-adic expansion $0 + 0 \cdot p + 0 \cdot p^2 + \ldots$ and so lies deep in the middle of the circles..." I'm not sure what you mean by this. If the blue disks stands for the residue class zero, then the element zero is the unique element lying in all of the blue disks. But terms like "deep" and "in the middle of" strike me as potentially misleading, because every element of $\mathbb{Z}_p$ has a similar description: no point is any deeper or more in the middle than any other point (and indeed the isometry group acts transitively). | |
| Jan 13, 2011 at 4:41 | history | answered | Daniel Litt | CC BY-SA 2.5 |