Cubic monogenic number fields that are non euclidean I'm looking for explicit examples of $\alpha$ such that $\mathfrak{O}_k = \mathbb{Z}[\alpha]$ for some cubic monogenic number field $k$ and where $\mathfrak{O}_k$ is not Euclidean. As much as possible, I'm hoping to find the simplest examples as I'm going to operate on their integer ring (as in add, multiply, modulo operations). I'm actually looking for simple monogenic fields of degree $\geq$ 3 with non-Euclidean integer rings and chose the cubic fields since I think computations would be simpler here.
 A: Every prime of $R=\Bbb{Z}[(15)^{1/3}]$ above $p\nmid 15$ is unramified and $(3,(15)^{1/3})^3=(3),(5,(15)^{1/3})^3=(5)$ whence every prime ideal is invertible and $R$ is the ring of integers of $\Bbb{Q}((15)^{1/3})$.
Its class number is not one because $I=(2,(15)^{1/3}-1)$ is not principal.
This follows from a numerical check: we have a unit $u = 1-30 (15)^{1/3}+12 (15)^{2/3}\in (0,1)$, if $I=(a)$ then replacing $a$ by $\pm au^n$ we can assume that $a\in (u,1)$ so that $2/a\in (2,2/u)$.
$a = c_0+c_1(15)^{1/3}+c_2(15)^{2/3}$, $\sigma(a)=c_0+e^{2i\pi/3}c_1(15)^{1/3}+e^{4i\pi/3}c_2(15)^{2/3}$, we have $N(a)=a \sigma(a)\overline{\sigma(a)}=2$ whence $a\in (u,1)$,$2/a\in (2,2/u)$ gives some bound $|a|, |\sigma(a)|,|\overline{\sigma(a)}|< \sqrt{2/u}$ from which we get some bounds for the integers $c_j$, so we can test them all.
I won't do the computation but it suffices to check that $\det(c_0+c_1\pmatrix{0&0&-15\\1&0&0\\0&1&0}+c_2\pmatrix{0&0&-15\\1&0&0\\0&1&0}^2)\ne  2$ for $|c_j|<100/\sqrt{u}$.
