On some cubic or quartic algebraic units

Let ? be an algebraic unit such that rank of the unit group of the order Z[?] is equal to one. It is natural to ask whether ? is a fundamental unit of this order. To prove this result, we showed that it suffices to find explicit positive constants c₁, c₂ and c₃ such that for any such ? it holds that c₁c₂|?|?d??c₃|?|^2c₂, where d? denotes the absolute value of the discriminant of ?, i.e. of the discriminant of its minimal polynomial. We give a proof of this result, simpler than the original ones.