Greetings.
I would love to have a field $\mathbb F$ which is a subfield of the field of rational numbers $\mathbb Q$, and such that the Galois group $Gal (\mathbb Q / \mathbb F)$ has preferably infinitely many elements.
While there is no such field $\mathbb F$, since $\mathbb Q$ has no proper subfields at all, I've recently heard of this field $\mathbb F_1$ with one element concept.
As far as I understand there is no definition which would be set in stone for this object, at least not yet. My question to those who know the subject: does any of the currently studied definitions of $\mathbb F_1$ allow for realization of $\mathbb F_1$ as a "subfield" of $\mathbb Q$ in some sense?
Best Answer
Imo the best theory today for the field with one element is Borger's proposal to consider Lambda-rings and use their Lambda-structure as a substitute for descent from the integers (or rationals) to the field F1 with one element.
Some examples of this philosophy are contained in the nice short paper by Borger and Bart de Smit (arXiv:0801.2352) 'Galois theory and integral models of Lambda-rings'.
Lambda-rings finite etale over the rationals Q are finite discrete sets equipped with a continuous action of the monoid Gal(Qbar/Q) x N' where N' are the positive integers under multiplication. This suggest that the Galois monoid Gal(Qbar/F1) = Gal(Qbar/Q) x N'.
Likewise, Lambda-rings over Q having an integral Lambda-model correspond to finite sets with a continuous action of the monoid Zhat, that is the set of profinite integers as a topological monoid under multiplication. This suggests that the absolute Galois monoid of F1, that is Gal(F1bar/F1) = Zhat.