On some cubic or quartic algebraic units

Let ϵ be an algebraic unit such that rank of the unit group of the order Z[ϵ] is equal to one. It is natural to ask whether ϵ 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 ϵ it holds that c1c2|ϵ|⩽dϵ⩽c3|ϵ|2c2, where dϵ denotes the absolute value of the discriminant of ϵ, i.e. of the discriminant of its minimal polynomial. We give a proof of this result, simpler than the original ones.