Why isn't there a structure with two primes?

Question

I don't know whether this question is a bit too vague for MO or not, so feel free to delete it if you see fit.

The p-adic integer is defined by taking the inverse limit $\ldots \mathbb{Z} / p^2 \rightarrow \mathbb{Z}/p $.One way to see the p-adic integers is to see it as dealing with $ \mathbb{Z} / p, \mathbb{Z} / p^2, \ldots $ at the same time. So $p$-adic integers allow us to see the structure of the ring of integers at the prime $p$. Taking the fractional field we obtain the $ p$-adic rational field $\mathbb{Q}_p$.

This construction is useful in study the arithmetic of the ring (field). For example, in the theory of class field theory, we study the question in $\mathbb{Q}_p$ first and glue them together and do something more to get the solution for $\mathbb{Q}$.

I want to ask why it is not possible for us to construct a structure that will allow us to see the ring of integer at two primes $p,q$ together and see how do they interacts? The analog of the above inverse limit construction seems to still work though not an intergral domain. However, we can still localize where it is possible.

Here is the motivation for the above question. We know that the global question are not solve by simply gluing together solution for local question. I attribute that to the fact that the primes does not play alone but interact with one another. An illustration of this can be seen through the fact that the product of all normalized absolute value is 1. So my question is why not isolate two primes to understand how they are interacting with one another instead of looking at all of them at once. I think there is some complicated issue that will arise from this. Just want to know what they are.

A more particular question may be like this: Let call the construction obtained above $\mathbb{Q}_{p,q} $. Is there something in the same vein of class field theory for this object. What is the obstacle in having such a theory. I am vaguely know that we have a more general Galois theory not only for fields but for rings also.

Answer 1

As stankewicz said, it is a general principle in number theory that whenever only finitely many primes are involved, they act "independently" in the sense that analyzing what is happening locally at each prime separately is enough to understand what all the finitely many primes are doing. One example of this is the Chinese Remainder Theorem.

Here is another: if you want a version of the integers with two primes $p$ and $q$, start with $\mathbb{Z}$ and invert every prime $\ell \not \in \{p,q\}$. This gives a semilocal ring with $(p)$ and $(q)$ as the nonzero primes. Similarly, you can do this with any two prime ideals $\mathfrak{p}, \mathfrak{q}$ in a Dedekind domain $R$. But the ring you get is not very interesting: it is a semi-local Dedekind domain, hence its class group is trivial, very likely the unit group $R^{\times}$ will be infinitely generated, etc. This domain is the intersection of the two DVRs $R_{\mathfrak{p}}$, $R_{\mathfrak{q}}$, and everything you want to know about it can be reduced to the DVRs. The same with two replaced by any finite set...

Answer 2

As mentioned in the comments, mimicking the construction of the $p$-adic numbers for two primes doesn't produce much new (although one might be led towards interesting stuff like $\widehat{\mathbb{Z}}$, the adeles, etc. - indeed, the 10-adics are sometimes used to motivate discussion of the $p$-adic numbers).

There is a rather speculative way of 'measuring' the arithmetic interaction between two primes based on absolute i.e. ${\mathbb F}_1$-derivations given in Kurokawa et al's paper Absolute Derivations and Zeta Functions. Here, one is looking at the (not necessarily additive) maps $D:{\mathbb Z}\rightarrow{\mathbb Z}$ satisfying $$D(ab)=D(a)b+aD(b).$$ These are essentially generated by the 'number derivative' for prime $p$, $$\frac{\partial}{\partial p}:mp^n\mapsto \begin{cases} mnp^{n-1}\qquad\mbox{for $n\geq 1$,} \\ 0\qquad\qquad\;\,\mbox{for $n=0$} \end{cases}$$ where $(m,p)=1$.

Now one can ask how two primes $p$ and $q$ are interacting by looking at the commutator of $\partial/\partial p$ and $\partial/\partial q$. In particular, in the paper above, they construct a zeta function out of this map, meromorphically continue it and look at the value at 0. They call this the quantum noncommutativity of $p$ and $q$, which is given by the bizarre-looking formula $$\frac{1}{12pq}\left((q-1)\sum_{k=1}^{\infty}\frac{p^kq^{p^{-k}}}{(1-q^{p^{-k}})^2} - (p-1)\sum_{k=1}^{\infty}\frac{q^kp^{q^{-k}}}{(1-p^{q^{-k}})^2}\right)$$ and somehow measures how $p$ and $q$ interact. What one might do with that is anybody's guess but it's certainly an interesting paper, recommended reading for anyone with an interest in ${\mathbb F}_1$-ideas.

Answer 3

This might be slightly off-topic, but I am surprised that no one mentioned Gauss' Quadratic Reciprocity: this is one of the very few fundamental results that allows you to deduce something about one prime, knowing something about a different prime. The intertwining is really surprising and (in some not so remote sense) it opens the way to Brauer-Manin obstructions. In fact, I am not sure that I know of any Brauer-Manin obstruction that may not eventually be reduced to an application of Quadratic Reciprocity.

EDIT: The reason for bringing up the Brauer-Manin obstruction is that it provides "equations" in the space of adelic solutions to a diophantine problem. These relations, while somehow "invisible" from each individual prime, become apparent by considering all the primes at once. Gauss' reciprocity then comes in when trying to make everything explicit and "squeezing" the information from all the primes to an interaction between only two primes. This is the relation to the two primes in the question and to the various inverse limits of Z considered in the other answers.

All of this was implicit in my comment, but I thought that it might be a good idea to expand in this.