Realizable classes of metacylic extensions of degree \(lm\)
Let \(O_k\) be the ring of integers of some number field \(k\), and \(\Gamma\) a metacyclic group of order \(lm\), where \(l\) is a rational prime and \(m\) some positive integer. Let \(\mathcal M\) be a maximal \(O_k\)-order in \(k[\Gamma]\), which contains \(O_k[\Gamma]\), and \(Cl (\mathcal M)\) the class group of locally free \(\mathcal M\)-modules. One is interested to study the subset \(\mathcal R (\mathcal M) \subset Cl (\mathcal M)\) of realisable classes, i.e. of those classes \(c\) such that there exists some tame Galois extension \(N/k\) with Galois group isomorphic to \(\Gamma\) and with \(\mathcal M \otimes_{O_k [\Gamma]} O_N\) lying in \(c\). It is conjectured that \(\mathcal R (\mathcal M)\) is even a subgroup of \(Cl (\mathcal M)\), which is true in the abelian case. In the present paper, the authors investigate the subset \(\mathcal R_1 (\mathcal M) \subset \mathcal R (\mathcal M)\) arising from those extensions \(N/k\) which are linearly disjoint from \(k(\zeta)/k\), where \(\zeta\) denotes an \(l\)-th root of unity. They use a Stickelberger ideal to give an explicit description of \(\mathcal R_1 (\mathcal M)\) and show that \(\mathcal R_1 (\mathcal M)\) is indeed a group.
scientific article; zbMATH DE number 5731637
scientific article