most recent 30 from http://mathoverflow.net 2013-05-23T03:22:26Z http://mathoverflow.net/feeds/question/86961 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/86961/invertible-elements-in-generalized-fields Martin Brandenburg 2012-01-29T15:03:17Z 2012-01-31T09:10:27Z

<p>Durov's <a href="http://arxiv.org/abs/0704.2030" rel="nofollow">theory</a> of generalized rings also includes generalized fields (5.7.6), which are defined as generalized rings, which are not subtrivial and whose proper strict quotients are subtrivial. For example, a classical ring is a generalized field iff it is a classical field. Nonclassical examples include $F_1$ ("the field with one element"), $F_{\infty}$ ("the residue field of the valuation ring of $\mathbb{R}$") and $F_{\emptyset}$ (the initial field).</p> <p>Now Durov mentions (6.1.16) that every generalized field $K$ embeds into its generalized ring of total fractions $K':=T^{-1} K$ and that $K'$ is a generalized field such that every non-zero element of $|K'|$ is invertible.</p> <p><strong>Question 1.</strong> Is there a generalized field $K$ such that $K \neq K'$, or in other words, such that not every non-zero element of $|K|$ is invertible?</p> <p>Durov mentions (5.7.9) that this is unclear. I think that there should be a counterexample, but not within the generalized fields mentioned above. </p> <p><strong>Question 2.</strong> Does every non-subtrivial generalized domain $A$ embed into a generalized field?</p> <p>Note that $A$ embeds into its total ring of fractions $A'$, which has again the property that every non-zero element of $|A'|$ is invertible, but it is not true in general that $A'$ is a generalized field ($A=\mathbb{F}_{\emptyset}[T]$ is easily seen to be a counterexample, here $|A| = \{T^n : n \in \mathbb{N}\}$ and $|T^{-1} A| = \{T^z : z \in \mathbb{Z}\}$ has lots of proper quotients).</p>