Im trying to figure out how i'd prove that the p-adic integers are the collection of $x \in Q_p so that |x| \leq 1$. There is a nice proof in Katok that shows that all p-adics less than or equal to 1 are p-adic integers, but it does not show that anything outside this "zone" is not a p-adic integer. Any help would be greatly appreciated, thanks in advance.
When I was taking graph theory, a certain theorem really caught my attention. It was due to Schur (I think) and the statement was that:
For all [;n;] and [;p;] a prime sufficiently large, then the equation [;x^n+y^n \equiv z^n;] mod [;p;] has non-zero solutions.
I'm not too familiar with [;p;]-adic integers but I was wondering, since its elements are just infinite lists of "consistent" residues mod [;p^k;], is there anyway to lift these non-trivial solutions to powers of primes and get a counter example to Fermat's Last Theorem on the [;p;]-adic integers?
Thanks in advance for the help!
I'm creating a presentation on an introduction to p-adic numbers, and I'm stuck. I want this presentation to be accessible to the general YouTube audience. I started with my current conceptual understanding of the archimedean property. I start by saying that 5/3 can be considered as 5 thirds (as in there are five of them). I'm pretty sure I get this luxury because of the archimedean property (please correct me if I'm wrong), which I interpret as the ability to "measure" object B with the "ruler" object A using an integer number of objects A. I used Wikipedia's visual on this property: https://en.wikipedia.org/wiki/Archimedean_property#/media/File:Archimedean_property.png
This is where things get weird. Ostrowski's theorem is a great way of deriving the other nontrivial absolute value, but this feels like we're grabbing an object out of convenient messy mathematics. I still don't have a RELATIVELY simple understanding of p-adic numbers beyond the proof of its special place in math.
Anyone have a resource that introduces the p-adic numbers in a much slower and productively conceptual way?
In the department I'm studying in, every week or two someone gives an informal talk on their research to other researchers and to the postgrad students -- to encourage students to come, the masters students have to write up a summary of the seminars that they have attended, and it counts as part of the graduation requirements.
The rules don't actually require that the seminars be at the university; if I see some interesting seminar somewhere else, there's great encouragement to go there. Given that so much is happening online these days, I can set my sights higher than just my local city.
For my thesis I'm looking at p-adic metrics as applied to machine learning; I'm much weaker in number theory than I am at machine learning, so I'm looking around for seminars that might stretch my knowledge of useful things that you can do with a p-adic metric.
Does your research group have a regular informal seminar mailing list that I could join? Or a website where I can see your seminars that I can lurk in?
So just got introduced to p-adic numbers. I understand that that rationals have holes which are filled by irrationals.
P-adic numbers provide another way to fill these holes. Why is this even needed?
Also can you link anything that formally builds up the p-adic numbers and shows applications?
for simplicity im only going to write numbers in base 10 and use the 10-adics in the post. when i write a number like "..123123" its just shorthand for "3*10^(0) + 2*10^(1) + 1*10^(2) + 3*10^(3) + 2*10^(4)..." etc.
a number like "..1212" diverges in the reals but converges in the 10-adics. a number like "0.1212.." converges in the reals but diverges in the 10-adics. a number like "..12125.1313.." should diverge both in the 10-adics and the reals, but doing arithmetic with numbers like that works totally fine.
>0.99.. = 1 (in the reals)
>
>..99 = -1 (in the 10-adics)
>
>..99 + 0.99.. = 1-1 -> 0 = ..99.99..
..99.99.. should diverge both in the 10-adics and the reals, but using this "hybrid system" it just represents 0. and we can go on..
>0/9 = ..99.99../9 -> 0 = ..11.11..
>
>n*0 = n*..11.11.. -> 0 = ..nn.nn.. (where n stands for an arbitrary digit between 0 and 9)
so ..44.44.. represent 0, ..33.33.. represent 0, and ofc ..00.00.. represents 0. and strange as the system looks, it seems totally consistent.
>1/3 = 0.33.. (in the reals)
>
>2/3 = ..334 (in the 10-adics)
>
>1/3+2/3 = 1 -> 0.33.. + ..334 = 1 -> ..334.33.. = 1
does this make sense? well recall that ..33.33.. = 0,
>..334.33.. - 33.33.. = 001.00 = 1 -> ..334.33.. = 1
you can also use this system as a shortcut to find 10-adic representation of numbers.
>4/9 = 0.444..
>
>0 - 4/9 = -4/9
>
>..44.44.. - 0.44.. = ..44 -> -4/9 = ..44
and yes this is true, ..44 is the 10-adic representation for -4/9.
>-4/9+1 = 5/9 -> ..445 = 5/9
etc..
i might also mention that you can show that any sequence of digits that repeat in both directions is 0, so ..1212.1212.. is 0, ..345345.345.. is 0, ..5757.6767.. is not 0 though because it's two different sequences.
so my questions are: what kind of number system is this? does it have a name? the p-adics can be rigorously defined using cauchy sequences or by using modular arithmetic, can the hybrid system be rigorously defined too?
(also i apologize if this post seems messy, tbh i find the "..." notation horrendous, i much more prefer representing repeating digits like this, but doing that on reddit isn't that easy)
I'm lost, I can't seem to see how the sequence 9, 99, 999, 9999, ... converges to 0 when you supply the integers with the 3-adic metric. Thanks.
When learning introductory real analysis, one often begins with exploring the structure of R, the reals. There are a number of different ways to construct the real field R: using Dedekind cuts of rationals, equivalence classes of Cauchy sequences of rationals, infinite decimal expansions, etc. Ultimately, though, the choice of construction doesn't matter in the following sense: any two complete ordered fields are unique, up to unique isomorphism. That is, R has an "intrinsic" characterization as the complete ordered field.
Now, in this context, I've been learning about the p-adic integers Z_p and the p-adic field Q_p. Analogous to the Dedekind cut versus Cauchy sequence constructions, there are different ways of constructing Z_p and Q_p. For example, one can realize Z_p as the completion of Z with respect to the p-adic metric. Additionally, Q_p can itself be constructed either directly as the p-adic metric completion of Q or as the field of fractions of Z_p.
Alternatively, one can construct Z_p as an inverse limit of surjective ring homomorphisms of the form Z/p^(k+1)Z → Z/p^(k)Z, where the homomorphisms are taken in the natural way. Further, one can topologize Z_p, this inverse limit, by giving each component Z/p^(k)Z the discrete topology. Under this topology, Z_p is metrizable relative to the p-adic metric.
Now, I'm aware that both R and each Q_p have unique field structures in the sense that, like R, each Q_p has a unique field automorphism. That suggests that, like R being the unique complete ordered field, there might be a similar intrinsic characterization of each Q_p and/or each Z_p.
Is there such an intrinsic characterization of the p-adics? If so, what is it? Does it also recover the metric and topological structure of the p-adics? (Such a characterization would be dependent on the prime p, of course, so first fix p.)
Thanks for any guidance!
I'm currently studying with Gouvea's book on p-adic numbers. I'm having a Problem with Problem 60. First let me define a few things: Let [; | \cdot | ;] a nonarchimedean absolute value on K. Than the valuation Ring is defindes as: [; \theta = \Bar{B}(0,1) =;] {[; x \in K: |x| \leq 1 ;]} The valuation ideal is defined as: [; \mathcal{B} = B(0,1) = ;]{[; x \in K: |x| < 1 ;]} And finally the residue field of [; | \cdot | ;] is defined as [; \mathcal{k} = \theta / \mathcal{B} ;]
In Problem 60 we are examining the Polynomial F(t) (F is a field). The Elements of F(t) are: [; \frac{f(t)}{g(t)} ;] with [; g(t) \neq 0 ;]. On F(t) we have the valuation: [; v_{\infty } = ( \frac{f(t)}{g(t)} )= deg(g(t)) - deg(f(t)) ;] With this valuation we get the non archimedean absolute value: [; | f(t) |{\infty} = e^{-v{\infty}(f(t))} ;]
I now want to compute [; \mathcal{k} = \theta / \mathcal{B} ;] for [; | \cdot |_{\infty} ;] My solution is : [; \mathcal{k} = { \frac{f(t)}{g(t)} \in F(t):deg(f) = deg(g) ;] My professor said that it can be shown that [; \mathcal{k};] is isomorph to F. I have no idea how to show that and am absolute stuck at that Problem. I need to present this Problem on Friday so this is kind of urgent. Thanks in advance
I am interested in p-adic geometry, and was wondering what some interesting open problems are. Things that I'd like to research in that area are as follows:
- Geometric Satake equivalence
- Selmer Groups
- Isogeny Classes
- Representations of Shtukas
Those topics are just there so that you have an idea on what I'd be interested in researching... But how could I apply these interests to the field of p-adic geometry?
I was just playing with the concept of p-adic numbers, which is new to me, and came up the following construction, that tries to unify p-adic topology and real topology in one. From the general properties of these topologies, I suspect that resulting topology must be trivial, but I can't see how it follows. The construction is:
Take cartesian product of the topological spaces of reals and p-adics (for some p): ℝ×ℚₚ. Then consider the relation ~ on this space: (x, a) ~ (x*q,a/q) for all nonzero rational q. It is clearly an equivalence relation.
Then take a quotient space by this equivalence relation: (ℝ×ℚₚ/~). What topology does it have? I have no idea.
If I am not wrong, there must be continuous functions both from ℝ and ℚₚ to this space:
ℝ→(ℝ×ℚₚ/~) : x↦[x,1]
ℚₚ→(ℝ×ℚₚ/~) : a↦[1,a]
(where [...] designate equivalence class by ~). Moreover, for rational arguments these functions agree, because [q,1] = [1,q] for rational q. Existence of such functions suggest that resulting topology must be trivial, but I am not sure.
edit: few typos, newlines.
edit2: change p to a in some places to avoid ambiguity
Hi, So I’ve been studying the p-adics, and as I understand them, they’re numbers that continue to the left instead of to the right, and use a prime number as a base. My question is, can you have a transcendental p-adic number? For example, ...876543210. Is this possible using p-adics, or am I misunderstanding p-adics?
Hello. My university is offering a class on p-adic Hodge theory, and I was wondering somethings about it. The description says that we will study p-divisible groups, isocrystals, Serre-Tate Theory Dienudonne-Manin Classification, and Hodge Tate decomposition.
Now, I do not really know what any of that is, but I am wondering why this stuff is useful. I guess I am particularly wondering if there are applications to diophantine equations.
I have taken a one semester class in algebraic number theory, and have not taken class field theory.
Here is my work for the integral done in tate's thesis. I get the euler factor that i desire just slightly shifted. I'm not sure what im doing wrong and need someone to help me. Thanks in advance for anything you can provide.
First, note that all rational numbers can be expressed as the average of two other rational numbers, i.e. for any x, there is a y,z such that x=(y+z)/2
This allows us to use induction to cover all rational numbers, starting with the integers and advancing to ever more precise rational numbers by repeated averaging
Given this insight, the proof that follows is rather trivial:
Base case: For x an integer, it can be expressed as x/2^0
Inductive case: Assume that y and z can be expressed as p/2^n and q/2^n respectively, with (y+z)/2 = x
Then x = (p/2^n + q/2^n )/2 = (p + q)/2^(n + 1), completing the induction
■
Unfortunately this proof is entirely non-constructive, while it's obvious that 1/3 can be expressed in the form p/2^n , it's not at all clear what values of p and n make the equation true. We just know that such integers exist and must leave it at that.
I'm looking for a good resource to start learning p-adic analysis. I am fairly comfortable with the concept of the p-adics and went through the analytic construction, but now I'd like to go deeper into it. My goal is to push myself to learn some modern areas of math where the p-adics are used in a practical way. I'm fairly comfortable with a lot of topics in math. I have completed courses in real analysis, functional analysis, complex analysis, algebra (groups, rings, fields, category theory, Galois theory), and a few number theory courses. Any help would be greatly appreciated, thanks in advance.
I recently learnt of p-adic numbers, and I think it's interesting, confusing, and potentially enriching...
It's 'confusing' to me, because it challenges some core assumptions that I've been using my entire life - eg. that numbers are close to each other if their difference is close to zero...
In any case, I've been thinking about it on-and-off for a few weeks, and I'm starting to get a good handle on it and how it works - but I'd be interested to learn more. In particular, I'd like to know if it is possible to do something similar to calculus with these numbers.
Does anyone know of any online course that I can look at that focuses on this topic?
So if I were supposed to name the video of 3Blue1brown with the most problems and most derserving of a revisit, it probably has to be "What does it feel like to invent math". There he tried to teach the viewers a little bit about how to do math and how new math can be found, with the example being p-adic numbers. Since these were only secondary, he tried to it without any actual math. The problem is that this lead to far more confusion. Most people didn't understand it and didn't realize what he wanted to tell them. The amount of comments trying to apply the real numbered concepts of |p|>1 is infinite.
in total, it is a shame since this is also one of my favourite videos, but it is practically impossible to understand if you don't already know the concepts it wants to teach you. Also because p-adic numbers are awesome.
So more clarificatio on this would be really awesome. It could be about metrics, sequences and convergence in general, but I'd also love another one about p-adic numbers. One way to do this could be with trees, which are verysimilar to your "rooms" but easier to understand. I recently stumbled upon this detailed article (www.colby.edu/math/faculty/Faculty_files/hollydir/Holly01.pdf) about how to visualize p-adic numbers via trees, and I really liked the idea, since it gives us a way to think about p-adic distances of rational numbers without automatically thinking them as an ordered line
I am doing a seminar in algebra and I'm currently learning about P-adic numbers.
and I don't understand this:
why every invertible P-adic number can be written as the product of P^n and an invertible P-adic integer?
http://www.ima.umn.edu/2018-2019/SW11.14-16.18/27655
Abstract:
"I will describe some of the history of, progress in, and future prospects for the p-adic Langlands program. This is an aspect of the Langlands program that grew out of the successful proof of Langlands reciprocity in various important cases (in particular, the modularity of elliptic curves over Q) twenty or so years ago. It relates the deformation theory of Galois representations to the representation-theoretic aspects of the theory of automorphic forms, for example via the investigation of representations of p-adic groups on p-adic vector spaces. As I will explain, while there has been significant progress in the p-adic Langlands program, a large amount remains to be done --- indeed, even the basic conjectural framework of the program remains unsettled. In the talk I hope to indicate some possibly fruitful directions for future research."
I have just stumbled on these, and I'm trying to wrap my head around them, any eli5 would be amazing too because I'm gobsmacked by this idea.
Hello!
Let the infinite series f(x) = 1 + x + x^2 + x^3 + ... and g(x) = 1 + 1/x + 1/x^2 +1/x^3 + ...
I was wondering, is there any value of x that would make f(x) and g(x) convergent? I did a bit of research and found something called "p-adic numbers", but I'm not sure what they are. Can someone explain p-adic numbers to me? I'm currently taking Calculus BC; do p-adic numbers tie into Calculus BC?
