Wikipedia: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)