On the minimal number of ramified primes in some solvable extensions of \(\mathbb Q\).
For a finite solvable group \(G\), let \(\text{ram}(G)\) (resp. \(\text{ram}^t(G))\) denote the minimal integer such that \(G\) can be realized as the Galois group of an extension (resp. a tamely ramified extension) over \(\mathbb{Q}\) ramified only at \(\text{ram}(G)\) (resp. \(\text{ram}^t(G)\)) finite primes. The paper, which is a part of the Ph. D. thesis of the author, is devoted to get bounds for \(\text{ram}(G)\) and \(\text{ram}^t(G)\) when \(G\) is either a finite nilpotent group of odd order or a generalized dihedral group. For instance, let \(l\) be an odd prime. We know by the Scholz-Reichardt's theorem that any \(l\)-group \(G\) can be realized as the Galois group of some extension over \(\mathbb{Q}\). By Burnside's basis theorem and Kronecker-Weber's theorem, \(\text{ram}(G)\) must be greater than or equal to \(d(G)\) the minimal number of generators of \(G\). It is not yet proved whether his lower bound coincides with \(\text{ram}(G)\) and \(\text{ram}^t(G)\) although this is claimed in [\textit{A. Cueto-Hernandez} and \textit{G. D. Villa-Salvador}, Pac. J. Math. 196, No.~2, 297--231 (2000; Zbl 0961.12003)]. In the paper under review, the author obtains that \(\text{ram}(G) = \text{ram}^t(G) = d(G)\) for every finite nilpotent group \(G\) of odd order satisfying a certain condition on \(d(G)\).
scientific article; zbMATH DE number 2101168
scientific article