TITLE: What is $\mathbb{Q}_1$, the "field of $1$-adic numbers"?
QUESTION [21 upvotes]: (Disclaimer: I'm totally ignorant about $\mathbb{F}_1$ theory)
There are now (several) working definitions of the "field with one element" $\mathbb{F}_1$ (not literally a field, of course), and certain mathematical objects "over $\mathbb{F}_1$" are recognized. For example, $\mathbb{F}_1$-vector spaces are pointed sets, $\mathrm{GL}_n(\mathbb{F}_1)$ is the symmetric group $\mathfrak{S}_n$ on $n$ elements (or was it $n-1$? $n+1$?). As far as I remember there's even a notion of "algebraic variety over $\mathbb{F}_1$": they are related to toric varieties.
So, my question:

Say $\mathbb{F}_p$ is to $\mathbb{Q}_p$ as $\mathbb{F}_1$ is to some object that we call $\mathbb{Q}_1$. How can $\mathbb{Q}_1$ be described explicitely in terms of more conventional mathematical objects?

The same question could be asked for $\mathbb{Z}_1$, $\mathbb{C}_1$, $\Omega_1$.

REPLY [7 votes]: Completions of a field are associated to places of that field. So for instance, $\mathbb Q(i)$ has two different completions of residue characteristic $5$. But $\mathbb F_1$ is not supposed to be a place of $\mathbb Q$. It is supposed to be the base field of $\mathbb Q$, which is different.