I had the cyclotomic thing in my mind a few hours before I posted, but then forgot it...

Since the field needs to have f^(-n)(a) for all a, it seems elementary that Q_p and F_p((T)) can't work, as well as any finite extension of either. R has other trivial reasons. The argument does not work for the field Q_p adjoined with all roots of p: even though the absolute value of nth-root(p) converges to 1, there is no convergent subsequence. I'm not sure how to treat infinite extensions, is there a way to get the argument back working? (or does it actually work still and I don't see it?)

Since non-archimedian local fields are just finite extensions, this is quite elementary, and the argument doesn't work for infinite extensions, which are the only interesting cases.

Why? I'm thinking of some kind of inverse limit of extensions of function fields over algebraic closure of finite field might occidentally work.

Thanks! This is real food for thought. Can you think of other structures than 0-manifolds, that include, say, C and R, and have an invariant like the geometric Euler characteristic, that make F_1, in a sense, algebraically closed, or maybe not?

I read intently as I could, and it still seems all answers hold back from saying any opinion how this or that should look. If I asked what GL(F_1) should be, or about F_1^n, I would get the normal answers that are already considered natural and normal. In the question above I dare the community to go beyond.

Thanks for the concise answer about the power of p dividing the subfields. As to the mentioned fields for which this fails, as I said, I am not interested in those - p is totally ramified in all these failures. Can you give a hint for as to how I might prove the result?

I don't understand what 123>132>... means. These are arrangements you bet on? This sounds like a very important question! But please make it clearer...