Sur la monogénéité de l'anneau des entiers de certains corps de rayon. (On the monogenicity of the ring of integers of certain ray fields)
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 [\textit{W. J. Ellison}, \textit{F. Ellison}, \textit{J. Pesek}, \textit{C. E. Stahl} and \textit{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 \textit{M. Mignotte} and \textit{M. Waldschmidt} [preprint, Université de Strasbourg, 1989] to obtain much smaller upper bounds for the magnitude of the solutions and reduce the calculations substantially.
nonexistence of integer points on an elliptic curve
scientific article; zbMATH DE number 4171049
scientific article