TITLE: What is the algebraic closure of the field with one element?
QUESTION [28 upvotes]: If doing geometry over $\mathbb F_p$ means also using its algebraic closure, it must be interesting to talk about the algebraic closure of $\mathbb F_1$ - the field with one element.
I saw that the finite extensions of $\mathbb F_1$ are considered as $\mu_n$, but an article by Connes et al says that it is unjustified to think of the direct limit of these.
In their paper, the group ring $\mathbb Q[\mathbb Q/\mathbb Z]$ appears a lot. Maybe it's one of $\mathbb Q/\mathbb Z$, $\mathbb Q[\mathbb Q/\mathbb Z]$, $\mathbb Z[\mathbb Q/\mathbb Z]$ ?
What is the algebraic closure of the field with one element?
And then, what is $\overline{\mathbb F_1} \otimes_{\mathbb F_1}\mathbb Z$? 
This seems like a very interesting question...

REPLY [34 votes]: The algebraic closure of F_1 is the group of all roots of unity, and, tensoring it with Z gives the integral group ring Z[mu_infty], or, if you prefer Z[Q/Z].
For a readable account (and for folklore references such as Kapranov-Smirnov) see Yu. I. Manin's "Cyclotomy and analytic geometry over F_1" (http://arxiv.org/abs/0809.1564).