Fundamental units for orders of unit rank 1 and generated by a unit
Volume 108 / 2016
Banach Center Publications 108 (2016), 173-189
MSC: Primary: 11R16, 11R27.
DOI: 10.4064/bc108-0-14
Abstract
Let $\varepsilon$ be an algebraic unit for which the rank of the group of units of the order ${\mathbb Z}[\varepsilon]$ is equal to $1$. Assume that $\varepsilon$ is not a complex root of unity. It is natural to wonder whether $\varepsilon$ is a fundamental unit of this order. It turns out that the answer is in general yes, and that a~fundamental unit of this order can be explicitly given (as an explicit polynomial in $\varepsilon$) in the rare cases when the answer is no. This paper is a self-contained exposition of the solution to this problem, solution which was up to now scattered in many papers in the literature.
