The class number one problem for the non-abelian normal CM-fields of degree 24 and~40 (Q2773319)

In [Invent. Math. 29, 275--286 (1975; Zbl 0299.12010)] \textit{A. M. Odlyzko} proved that there are only finitely many normal CM-fields of class number one. This paper determines all such non-abelian fields of degree 24 and 40. Only three fields exist in degree 24 satisfying these requirements, while only one is of degree 40. More general results concern fields of degree \(8p\), \(p\) an odd prime. The proof relies on algebraic arguments combined with a lower bound for \(h^-_K\) given in \textit{S. Louboutin} and \textit{Y.-P. Park} [Publ. Math. 57, 283--295 (2000; Zbl 0963.11065)]. This latter provides an upper bound for the discriminant, and numerical computations enable the author to conclude his results. The key point is of course that the theoretical considerations have reduced the required computations to a sensible amount. Note that ``greater than'' is to be read as ``strictly greater than''.