On the fundamental units of a totally real cubic order generated by a unit

Abstract:

We give a new and short proof of J. Beers, D. Henshaw, C. McCall, S. Mulay and M. Spindler following a recent result: if $\epsilon$ is a totally real cubic algebraic unit, then there exists a unit $\eta \in \textbf {Z}[\epsilon ]$ such that $\{\epsilon ,\eta \}$ is a system of fundamental units of the group $U_\epsilon$ of the units of the cubic order $\textbf {Z}[\epsilon ]$, except for an infinite family for which $\epsilon$ is a square in $\textbf {Z}[\epsilon ]$ and one sporadic exception. Not only is our proof shorter, but it enables us to prove a new result: if the conjugates $\epsilon ’$ and $\epsilon ''$ of $\epsilon$ are in $\textbf {Z}[\epsilon ]$, then the subgroup generated by $\epsilon$ and $\epsilon ’$ is of bounded index in $U_\epsilon$, and if $\epsilon >1>\vert \epsilon ’ \vert \geq \vert \epsilon '' \vert >0$ and if $\epsilon ’$ and $\epsilon ''$ are of opposite sign, then $\{\epsilon ,\epsilon ’\}$ is a system of fundamental units of $U_\epsilon$.