Wikipedia:Reference desk/Archives/Mathematics/2018 July 8

= July 8 =

$$\mathbb{Z}[\sqrt{2}]$$ as a localization
In the principal ideal domain $$\mathbb{Z}[\sqrt{2}]$$, the element $$1 + \sqrt{2}$$ is a unit of infinite multiplicative order. Does there exist a subring R of $$\mathbb{Z}[\sqrt{2}]$$ containing $$1 + \sqrt{2}$$ but not its inverse $$-1 + \sqrt{2}$$ such that $$\mathbb{Z}[\sqrt{2}]$$ is isomorphic to the localization of R at the multiplicative set of powers of $$1 + \sqrt{2}$$ as R-algebras? If so, will R also be a PID, or at least a UFD? GeoffreyT2000 (talk) 20:00, 8 July 2018 (UTC)


 * Any subring of $$\mathbb{Z}[\sqrt{2}]$$ which contains $$1 + \sqrt{2}$$ must also contain $$(1 + \sqrt{2})^2 = 3 + 2\sqrt{2},$$ and thus must also contain $$5(1 + \sqrt{2}) - 2(3 + 2\sqrt{2}) = -1 + \sqrt{2},$$ which renders the rest of your question moot. –Deacon Vorbis (carbon &bull; videos) 20:47, 8 July 2018 (UTC)