Axiomatic Approach to p-adic rationals

Question

There are several ways to construct the real numbers: completion of the rationals w.r.t. the euclidean metric, dedekind completion of the rationals, infinite continued fractions, etc. Each of these constructions give different insights into the structure of the real numbers and how they relate to the rationals.

However, I have always best understood the real numbers to be the unique linearly-ordered field which is dedekind-complete. And each of the foregoing constructions can be demonstrated to give rise to such a structure (and thus we really are working with the same thing each time). I feel like this understanding is easiest for me because I'm not relying on the `representation' of a real number (i.e. a cauchy sequence of rationals, dedekind cuts, or infinite sequences of integers) to understand the real numbers.

Right now I find myself studying the $p$-adic rationals, and I've hit a stumbling block. As in the real case, there are a couple different ways to 'construct' $\mathbb{Q}_p$: first as a Cauchy completion of the rationals under the $p$-adic metric, or as $\mathbb{Z}_p[1/p]$ where $\mathbb{Z}_p$ is realized as the inverse limit of the well-known inverse system. I was initially surprised that the latter construction is purely algebraic in nature which can be equipped with a topology as an afterthought, and that the former construction is almost entirely topologically focused (the real construction always has either a metric or an ordering underlying the construction).

However, even with the benefit of having two different constructions in hand, I feel no more `enlightened' about what I'm working with because I'm relying on representations. I was wondering if anyone knew a few 'axioms' that uniquely identified $\mathbb{Q}_p$?

As the axiom system for $\mathbb{R}$ includes an ordering, I wouldn't have qualms about something like "ultrametric valued field" being incorporated into the axiom list.

Answer 1

Theorem: Every locally compact (Hausdorff) field whose topology is not discrete is a finite extension of $\mathbb{R}$, of $\mathbb{Q}_p$ for some $p$, or of $\mathbb{F}_p[[t]]$ for some $p$.

These are the local fields.

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Here is a completely different answer. To my mind, $\mathbb{Q}_p$ is best understood as the fraction field of $\mathbb{Z}_p$ (in the same way that $\mathbb{Q}$ is best understood as the fraction field of $\mathbb{Z}$), so this is really a question about $\mathbb{Z}_p$. Here is a definition of $\mathbb{Z}_p$ which at least has the benefit of not being a construction:

Let $A$ be an abelian $p$-group, meaning that every element of $A$ is $p$-power torsion. The natural action of $\mathbb{Z}$ on $A$ then naturally extends to an action of $\mathbb{Z}_p$: that is, it's meaningful to write $na$ where $n \in \mathbb{Z}_p$ and $a \in A$. In fact I claim that every "natural" endomorphism $A \to A$ has this form: that is,

The ring of natural endomorphisms of the forgetful functor from abelian $p$-groups to sets is precisely $\mathbb{Z}_p$.

For some guidance on how to prove results of this form see this blog post and this blog post.

A more down-to-earth definition involving a particular abelian $p$-group is that $\mathbb{Z}_p$ is the endomorphism ring of the Prufer p-group $\mathbb{Z}[1/p]/\mathbb{Z}$.

Answer 2

Following a lead given by Lubin, we have the following result:

$\Bbb Q_p$ is the unique local field such that

• $\overline{\Bbb Z}$ is an open subset
• $1/p\not\in\overline{\Bbb Z}$

where we have identified $\Bbb Z\subseteq \Bbb Q_p$ as the image of the unique ring morphism (preserving 1) from $\Bbb Z$ to $\Bbb Q_p$.

Exploiting the characterization of local fields, stated by Qiaochu, the first condition eliminates $\Bbb R$, $\Bbb C$, and $\Bbb F_p((t))$. Thus we need only consider finite extensions $K$ of $\Bbb Q_q$ for some prime $q$. However, if $[K:\Bbb Q_q]=n>1$, we clearly have that $\overline{\Bbb Z}$ is still not open as the topology on $K$ is homeomorphic to $\Bbb Q_p^n$ and $\overline{\Bbb Z}$ is contained in a one-dimensional subspace. This forces us to focus on $\Bbb Q_q$ for various prime $q$. Here, the second condition forces the local field to be $\Bbb Q_p$.