Abstract
Let e be an algebraic unit for which the rank of the group of units of the order ℤ[e] is equal to 1. Assume that e is not a complex root of unity. It is natural to wonder whether e 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 e) 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.
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