Norm-Euclideanity of the integers adjoined square root two
= Norm-Euclideanity of the integers adjoined square root two
{title2=$|N(a+b\sqrt2)|=|a^2-2b^2|$}
The ring $\mathbb Z[\sqrt2]$ is a <Euclidean domain> for the absolute <field norm>. Approximate both rational coefficients of a quotient by nearest integers. If the errors are $u,w\in[-1/2,1/2]$, then $|u^2-2w^2|\leq1/2<1$, giving a remainder of strictly smaller absolute norm. In particular this ring is a <principal ideal domain> and has trivial <ideal class group>.