The authors show that the main conjecture of monogenicity (see the previous review) is not true in general. In particular, let \(k={\mathbb{Q}}(\sqrt{-19})\), \(\omega =(1+\sqrt{-19})/2\). The ideal 7 splits in k/\({\mathbb{Q}}\) and \(P_ 7\), the ideal generated by \(q=1+\omega\), is one of the ideals dividing it. Then the ring of integers of the extension K of k of ray \(P_ 7\) does not possess a power basis over \({\mathbb{Z}}_ k\). To prove this the authors reduce the problem to showing the nonexistence of integer points of \({\mathbb{Z}}_ k\) on an elliptic curve and use the method of Baker and Ellison [W. J. Ellison, F. Ellison, J. Pesek, C. E. Stahl and D. S. Stall, J. Number Theory 4, 107-117 (1972; Zbl 0236.10010)] to resolve the problem. Alternatively, the authors show that they could have used the latest results of M. Mignotte and M. Waldschmidt [preprint, Université de Strasbourg, 1989] to obtain much smaller upper bounds for the magnitude of the solutions and reduce the calculations substantially.