Let ε be an algebraic unit for which the rank of the group of units of the order ℤ [ ε ] is equal to 1 . Assume that ε is not a complex root of unity. It is natural to wonder whether ε is a fundamental unit of this order. It turns out that the answer is in general positive, and that a fundamental unit of this order can be explicitly given (as an explicit polynomial in ε ) in the rare cases when the answer is negative. 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. We also include the state of the art in the case that the rank of the group of units of the order ℤ [ ε ] is greater than 1 when now one wonders whether the set { ε } can be completed in a system of fundamental units of the order ℤ [ ε ] .