Talk:Euclidean domains

Real quadratic field

The ring of integers of a real quadratic field with class number 1 and discriminant < 500 is proved to be Euclidean; see Section 5.2, Page 34, Theorem 6 of here. Of course this concerns $\mathbb {Z} [{\sqrt {14}}],\mathbb {Z} [{\sqrt {22}}],\mathbb {Z} [{\sqrt {23}}],\cdots$, which includes all real quadratic fields with status "???" in the table.

Note still that $\mathbb {Z} \left[{\dfrac {1+{\sqrt {69}}}{2}}\right]$ is current the only case where an explicit Euclidean function is constructed; for other rings in the family we just say that the "minimal Euclidean function" is well-defined. - Jianing Song (talk) 06:30, 24 November 2024 (EST)