On the fundamental units of some cubic orders generated by units
Volume 165 / 2014
Acta Arithmetica 165 (2014), 283-299
MSC: Primary 11R16; Secondary 11R27.
DOI: 10.4064/aa165-3-7
Abstract
Let $\epsilon $ be a totally real cubic algebraic unit. Assume that the cubic number field ${\mathbb Q}(\epsilon )$ is Galois. Let $\epsilon $, $\epsilon '$ and $\epsilon ''$ be the three real conjugates of $\epsilon $. We tackle the problem of whether $\{\epsilon ,\epsilon '\}$ is a system of fundamental units of the cubic order ${\mathbb Z}[\epsilon ,\epsilon ',\epsilon '']$. Given two units of a totally real cubic order, we explain how one can prove that they form a system of fundamental units of this order. Several explicit families of totally real cubic orders defined by parametrized families of cubic polynomials are considered. We also improve upon and correct several previous results in the literature.
