Visualizing $\textbf{Q}_p$ vs. $\textbf{F}_p((t))$?

Question

I have a few questions.

1. How do I visualize the field $\textbf{Q}_p$ of $p$-adic numbers?
2. How do I visualize the field $\textbf{F}_p((t))$ of Laurent series of $\textbf{F}_p$?
3. How do I do 1 and 2 in a way as to make the similarities and differences between $\textbf{Q}_p$ and $\textbf{F}_q((t))$ as transparent as possible?

Hopefully this question can be answered in a way that does not go too much off the deep end. I just want to visualize things!

EDIT: I want pictures.

Answer 1

• $\mathbf{Z}_p$ has two equivalent definitions : as $\varprojlim \mathbf{Z}/p^n \mathbf{Z}$ ie. the set of sequences $a = (a_1,a_2,\ldots), a_n \in \mathbf{Z}/p^n \mathbf{Z}, a_n \equiv a_m \bmod p^m \ (m \le n)$ with the pointwise addition and multiplication.

  Or as $p$-adic series, ie. the completion of $\mathbf{Z}$ for the (non-archimedian) $p$-adic absolute value $|k|_p = p^{-n}$ if $p^n | k, p^{n+1} \nmid k$.

  The $p$-adic series representation of $(a_1,a_2,\ldots)$ is $a_1+\sum_{n=1}^\infty b_n p^n$ where $b_n = \frac{a_{n+1}-a_{n}}{p^{n}}$

• $\mathbf{F}_p[[t]]$ is the completion of $\mathbf{F}_p[t]$ for the (non-archimedian) absolute value $|\sum_{l=m}^d a_l t^l|_v = p^{-m}$ if $a_m \not \equiv 0 \bmod p$

• $ \mathbf{Q}_p= \mathbf{Z}_p[p^{-1}]$ and $ \mathbf{F}_p((t))= \mathbf{F}_p[[t]] \, [t^{-1}]$

  An important difference is that $\mathbf{F}_p[[t]]$ is of characteristic $p$ (ie. $p = 0$) so you can't embed $\mathbf{Z}$ into $\mathbf{F}_p[[t]]$.

  But $\mathbf{Z}_p$ is of characteristic zero (completion of $\mathbf{Z}$ or inverse limit of rings of characteristic $p^n$)

Answer 2

Let's visualize the fragments $\mathbf Z_p$ and $\mathbf F_p[[t]]$ instead, since they're compact, and focus only on their structure as abelian topological groups. Then the left figure is $\mathbf Z_3$ and the right figure is $\mathbf F_3[[t]]$:

As you can see, they are homeomorphic as topological spaces, but not when you include the group structure. The greater visual regularity on the right figure is caused by the torsion in $\mathbf F_3[[t]]$: no matter how you shuffle it around, performing the same operation $3$ times gets you back to the identity. By contrast, the addition operation on $\mathbf Z_3$ is twisted enough to avoid that. If you rotate the level-$1$ balls three times, it amounts to a rotation of the level-$2$ balls, and so on. This is just a fancy way of observing that in base three, $1_3+1_3+1_3=10_3$.

Here's the mathematical justification for my hand-waving. Each group $G$ is visualized by an embedding $i:G\to\mathbb R^2$ in such a way that the group structure on $G$ is maximally respected by the differentiable structure on $\mathbb R^2$. To be precise, there exists a dense subgroup $H<G$ such that for all $x\in H$, the addition map $\sigma_x: G\to G, \sigma_x(y)=x+y$ extends to a differentiable map on a neighborhood of the image $i(G)$. This requirement is enough to force the two groups to embed into dissimilar subsets of $\mathbb R^2$.

Specifically, the left embedding is the Chistyakov embedding, $i(x)=\sum_{n=1}^\infty s^n \exp(2\pi i x/3^n)$ with an appropriate scale factor $0<s<1$. Some further discussion, with links to the literature (DOI:10.1007/BF02073866), is in this Wikimedia image description. The right embedding is just the usual $i(\sum_{n=0}^\infty a_n t^n)=\sum_{n=0}^\infty s^n \exp(2\pi i a_n/3)$.

(You will sometimes see something like the right figure being used to visualize $\mathbf Z_3$ instead. Just be aware that while it's a fine visualization of the topological structure, it does violence to the group structure.)

I intend to publish an article that explains the difference between these embeddings in greater detail, including the differentiability condition... someday. For now, enjoy the pictures!

Answer 3

You’ve contacted me directly, perhaps guessing (correctly) that I missed this question. Before I explain my general lack of a picture for the $p$-adics and the rings $\Bbb F_q[[t]]$, maybe I should point out a commonly-drawn picture that neither of the other answerers has mentioned.

It’s the graph-theoretical description of $\Bbb Z_p$, with an easy extension to a description of $\Bbb Q_p$. I’m afraid I’m too lazy to make up a picture, necessarily partial, of the graph. So you’ll have to draw it in your mind. Start with the root of the graph, a single vertex, up top, level $0$. Then on level $1$, one level below that, draw $p$ vertices, all connected to the root. Now, from each of these $p$ vertices, draw $p$ edges going down to $p$ more vertices, giving you $p^2$ vertices on level $2$, $p^2$ vertices in all. And continue infinitely.

The $p$-adic numbers, elements of $\Bbb Z_p$, are the ends of this graph. That is, each path leading from the root in level $0$ that goes all the way down, gives you a $p$-adic number. For two different elements, say $u,w\in\Bbb Z_p$, that is, two different ends of the graph, you get the $p$-adic distance between them by seeing how far up on the graph you need to go before you start going down to the other end. If you needed to go to level $2$, for instance, then $v_p(u-w)=2$, in other words $|u-w|_p=1/p^2$.

I said in my private e-mail to you that I really don’t have a picture. I do think of the “open unit disk” in the algebraic closure of $\Bbb Q_p$, that is, all $z$ with $|z|_p<1$, in other words with $v_p(z)>0$. But the only “picture” I have of it is completely unrealistic, as a seamless disk like the complex unit disk. But these pictures are important: a bad mental image can really throw you off, lead you astray seriously.

I like to keep in mind the difference between the complex and the $p$-adic open unit disks. The complex disk is connected but can’t be made into a group, while the $p$-adic open disk $D=D_{\text{$p$-adic}}$may be disconnected, but has the natural structure (via addition) of a group. Furthermore, you can put infinitely more completely different group structures on $D$. For instance you can translate the natural multiplicative structure of $1+D$ over to $D$ itself by the formula $x\star y=x+y+xy$. The wildly different other group structures need power series for their formula, but that’s all right…

Answer 4

Late answer, but I asked myself the same question and ended up trying to illustrate properties of the p-adic integers using the Chistyakov embedding.

The 3-adic world was the simplest one, where the following property : $$ord_p(a+b)≥\min(ord_p(a),ord_p(b))$$

can be seen as moving elements to the right. However, other properties (convergence, or visualisation for other values of $p$ were less successful).

More information and code can be found here: P-adic numbers visualization