On unit solutions of the equation \(xyz=x+y+z\) in totally imaginary quartic fields (Q1183270)

The equations \(xyz=x+y+z=1\) have been shown by J. W. S. Cassels to have no solutions in the rational number field \(\mathbb Q\). It is natural to ask when \[ u_ 1u_ 2u_ 3=u_ 1+u_ 2+u_ 3\tag{*} \] has a solution in the ring of integers \({\mathcal O}_ K\) of an arbitrary algebraic number field \(K\), where the \(u_ i\)'s are units in \({\mathcal O}_ K\). The reviewer, \textit{C. Small}, \textit{K. Varadarajan} and \textit{P. G. Walsh} [Acta Arith. 48, 341--345 (1987; Zbl 0576.10009)] completely solved the problem and gave a list of solutions for the case where \(K\) is a quadratic extension of \(\mathbb Q\). In earlier work the last two authors of the paper under review showed that \((*)\) has no solutions to any real number field with unit group of rank 1 and fundamental unit \(\eta>1\). Also in earlier work they showed that there are only 4 not totally real cubic fields in which \((*)\) is solvable. Together with H. Edgar they now consider the final case for which the rank of the group of units of the ring of integers \(K\) is 1; i.e., they determine when \((*)\) has a solution in a totally imaginary quartic extension of \(\mathbb Q\).