If doing geometry over Fp $\mathbb F_p$ means also using its algebraic closure, it must be interesting to talk about the algebraic closure of F_1 $\mathbb F_1$ - the field with one element.
I saw that the finite extensions of F_1 $\mathbb F_1$ are considered as mu_n, $\mu_n$, 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 Q[Q/Z] $\mathbb Q[\mathbb Q/\mathbb Z]$ appears a lot. Maybe it's one of Q/Z$\mathbb Q/\mathbb Z$, Q[Q/Z]$\mathbb Q[\mathbb Q/\mathbb Z]$, Z[Q/Z] $\mathbb Z[\mathbb Q/\mathbb Z]$ ?
What is the algebraic closure of the field with one element?
And then, what is Closure(F_1)xZ? (where the tensor is over $\overline{\mathbb F_1) } \otimes_{\mathbb F_1}\mathbb Z$? This seems like a very interesting question...
