The ring $\mathbb{Z}[2\cos(\frac{\pi}{k})]$ is known to be a Euclidean domain for $k=3,4,5$ and $6$, because in those cases $2\cos(\frac{\pi}{k}) = 1, \sqrt{2},$ the golden ratio $\phi$, and $\sqrt{3}$ respectively, and $\mathbb{Z}, \mathbb{Z}[\sqrt{2}], \mathbb{Z}[\phi]$, and $\mathbb{Z}[\sqrt{3}]$ are all known to be Euclidean domains.

I haven't been able to find a reference that will tell me whether $\mathbb{Z}[2\cos(\frac{\pi}{k})]$ is (or isn't) a Euclidean domain for integers $k$ higher than $6$, though. Is this an open problem?