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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...

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# What is the algebraic closure of the field with one element?

If doing geometry over Fp means also using its algebraic closure, it must be interesting to talk about the algebraic closure of F_1 - the field with one element.

I saw that the finite extensions of 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 Q[Q/Z] appears a lot. Maybe it's one of Q/Z, Q[Q/Z], Z[Q/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 F_1) This seems like a very interesting question...