Rational numbers as an extension of the field with one element? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T04:30:06Z http://mathoverflow.net/feeds/question/19131 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/19131/rational-numbers-as-an-extension-of-the-field-with-one-element Rational numbers as an extension of the field with one element? Ćukasz Grabowski 2010-03-23T18:43:05Z 2010-03-23T20:03:13Z <p>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.</p> <p>While there is no such field \$\mathbb F\$, since \$\mathbb Q\$ has no proper subfields at all, I've recently heard of this <em>field</em> \$\mathbb F_1\$ <em>with one element</em> concept.</p> <p>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?</p> http://mathoverflow.net/questions/19131/rational-numbers-as-an-extension-of-the-field-with-one-element/19139#19139 Answer by lieven lebruyn for Rational numbers as an extension of the field with one element? lieven lebruyn 2010-03-23T19:38:25Z 2010-03-23T20:03:13Z <p>Imo the best theory today for the field with one element is <a href="http://arxiv.org/abs/0906.3146" rel="nofollow">Borger's proposal</a> 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.</p> <p>Some examples of this philosophy are contained in the nice short paper by Borger and Bart de Smit (arXiv:0801.2352) <a href="http://arxiv.org/abs/0801.2352" rel="nofollow">'Galois theory and integral models of Lambda-rings'</a>.</p> <p>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'.</p> <p>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.</p>