Realizable classes by non-abelian metacyclic extensions and Stickelberger elements
A theorem of McCulloh (part of which was shown earlier by Childs) states that for a group \(\Gamma\) of prime order and a number field \(k\), the classes in \(Cl({\mathcal O}_k[\Gamma])\) arising as \([{\mathcal O}_N]\) with \(N/k\) a tame \(\Gamma\)-extension are exactly those which are in \(J\cdot Cl({\mathcal O}_k[\Gamma])\), where \(J\subset {\mathbb{Z}}[Aut(\Gamma)]\) is a suitably defined Stickelberger ideal. The paper under review studies the simplest case of a non-commutative Galois group: \(\Gamma\) is now supposed to be metacyclic non-abelian of order \(lq\), where \(l\) and \(q\) are prime numbers and \(q\) divides \(l-1\). The author proves a very nice result which determines the classes which are realizable by tame \(\Gamma\)-extensions \(N/k\), with the following change of scenario: one works in the class group of a maximal order \(\mathcal M\) in \(k[\Gamma]\), instead of the class group of \({\mathcal O}_k[\Gamma]\) itself. The result extends McCulloh's result in a satisfactory way: again Stickelberger ideals are the clue to the solution. One can get started since the class group of the above mentioned maximal order is easily obtained by Morita theory once one knows the decomposition of the group algebra \(k[\Gamma]\) into simple components. In fact, \(Cl({\mathcal M})\) is the product of the three class groups \(Cl({\mathcal O}_k)\), \(Cl({\mathcal O}_{k(\zeta_q)})\), and \(Cl({\mathcal O}_K)\), where \(K\) is the unique field between \(k\) and \(k(\zeta_l)\) such that \([k(\zeta_l):K]=q\). The proof of the main result uses the results for the abelian prime order case in a neat and efficient way. This is a line of research which deserves to be continued.
scientific article; zbMATH DE number 1037293