Let $\epsilon$ be an algebraic unit of the degree $n\ge 3$. Assume that the extension $\mathbb{Q}\left(\epsilon \right)/\mathbb{Q}$ is Galois. We would like to determine when the order $\mathbb{Z}\left[\epsilon \right]$ of $\mathbb{Q}\left(\epsilon \right)$ is $\mathrm{Gal}\left(\mathbb{Q}\right(\epsilon )/\mathbb{Q})$-invariant, i.e. when the $n$ complex conjugates ${\epsilon }_1,\cdots ,{\epsilon }_n$ of $\epsilon$ are in $\mathbb{Z}\left[\epsilon \right]$, which amounts to asking that $\mathbb{Z}[{\epsilon }_1,\cdots ,{\epsilon }_n]=\mathbb{Z}\left[\epsilon \right]$, i.e., that these two orders of $\mathbb{Q}\left(\epsilon \right)$ have the same discriminant. This problem has been solved only for $n=3$ by using an explicit formula for the discriminant of the order $\mathbb{Z}[{\epsilon }_1,{\epsilon }_2,{\epsilon }_3]$. However, there is no known similar formula for $n>3$. In the present paper, we put forward and motivate three...

In this article we will describe a surprising observation that occurred in the construction of quadratic unramified extensions of a family of pure cubic number fields. Attempting to find an explanation will lead us on a magical mystery tour through the land of pure cubic number fields, Hilbert class fields, and elliptic curves.