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

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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 • videos) 20:47, 8 July 2018 (UTC)