Given \(\beta> 1\), the greedy \(\beta\)-expansion of a number \(x\in [0; 1)\) is the expansion of the form \(x =\sum_{i\leq k}x_i\beta^i\), where \(x_i\in \{a\in\mathbb Z:0\leq a<\beta\}\) are given by \(T:[0,1)\mapsto [0,1)\), \(T(x) = \beta x - [\beta x]\). This can be extended to all \(x > 0\) by multiplying \(x\) with a suitable power of \(\beta\), and to negative numbers using the - sign. The set of numbers \(x\) for which such expansion is finite is denoted Fin(\(\beta\)). The authors study quadratic and cubic number fields admitting a Pisot unit \(\beta\) such that \(\mathbb Z[\beta,\beta^{-1}] \subseteq \) Fin(\(\beta\)) a property which they refer to as (F). For example, they show that in every real cubic field which is not totally real one can find a Pisot unit with the property (F), whereas there exist totally real cubic number fields without such a unit. The methods are in principle elementary (estimates for roots of certain cubic polynomials) except for a result of S. Akiyama [in: Algebraic number theory and diophantine analysis. Proceedings of the international conference, Graz, Austria, August 30–September 5, 1998. Berlin: Walter de Gruyter. 11–26 (2000; Zbl 1001.11038)] which describes the minimal polynomials of cubic Pisot units with the property (F).