p-adic integers: analytic properties (idea)

If you've read about p-adic integers (and maybe even if you haven't...) you'll know about the explicit representation of p-adic integers, as well as how to perform most types of arithmetic on them.

But p-adic integers are also the completion of the integers (or the natural numbers) in the p-adic norm. So we can perform analysis on p-adics, just like we can perform analysis on real numbers. Of course, our "usual" metric of the absolute value |.| isn't defined on p-adics, so we can't use it. Instead, we use the p-adic norm |.|_p.

Most of the results in this writeup may be easily extended also apply to p-adic rationals; when they don't, I'll try to note it.

What's the p-adic valuation of a p-adic?

Take some p-adic d=...d₂d₁d₀. If d were a natural number (i.e. all but finitely many d_i's were 0) then v_p(d) would be min { i : d_i != 0 }. So the same is true for any p-adic.

Recall that the distance between 2 p-adics x,y is |x-y|_p = p^-v_p(x-y). What this means is that p-adics are "backwards": x is close to y if they share many "low" digits.

Every p-adic integer is a limit of natural numbers

Since the p-adic integers are the completion of the natural numbers, this is obvious. Indeed, we see that the sequence of (base-p) natural numbers d₀, d₁d₀, d₂d₁d₀, ... converges to d (every element shares another digit with d).

In particular, we see that if x is a natural number, we may write down -x as a limit of natural numbers; there is no concept of a "positive" p-adic integer!

|.|_p yields an ultrametric

Ultrametrics are preserved in the completion; alternatively, this can be shown directly: the number of low digits shared by x and z is at least the smaller of the number of low digits shared by x and y and the number of low digits shared by y and z, for any y.

Z_p is compact

(This is not true of Q_p)

Students of analysis should already have recognised the topology given by the p-adic norm: it's the (infinite) Tychonoff product {0,...,p-1}×{0,...,p-1)×(0,...,p-1)×... By Tychonoff's theorem, it is compact.

Or we can do it directly. Say we have a sequence of p-adics

a₁=...a_1,2a_1,1a_1,0
a₂=...a_2,2a_2,1a_2,0
a₃=...a_3,2a_3,1a_3,0
...

Then infinitely many of the a's share the same 1's digit; pick a_n1 to be the first such. Of these, infinitely many share the same p's digit; pick a_n2 to be the first of these after a_n1. Continuing in this fashion, we build a sequence {a_nk}_k such that a_nj share the same left k digits for all j>k.

This sequence clearly converges, so we see Z_p is a compact metric space.

Z_p has cardinality aleph

As a perfect set that is dense in itself, it must be.

But we can see it directly. Any p-adic integer is, after all, an infinite string of the digits {0,...,p-1} that just happens to go left when we write it down. Basic set theory shows that the cardinality of the set of such strings (at least when they go to the right) is aleph.

Z_p is a totally disconnected set

(So it's what's often loosely called a Cantor set, and if p=2 then it has the same topology.

This follows directly from being the infinite product {1,...,p} × {1,...,p} × ...