Realizable classes of tetrahedral extensions
Given a number field \(k\) and a finite group \(\Gamma\), denote by \({\mathfrak M}\) a maximal \({\mathfrak o}_k\)-order in \(k[\Gamma]\) containing \(\Gamma\) and by \({\mathcal R}({\mathfrak M})\) the set of classes \(c\) in the class group \(\text{Cl}({\mathfrak M})\) such that there is a tame Galois extension \(K/k\) with \(G_{K/k}\simeq \Gamma\) and \(c=\) class of \({\mathfrak M}\otimes_{{\mathfrak o}_k[\Gamma]} {\mathfrak o}_K\). If \(\Gamma\) is Abelian, then \textit{L. R. McCulloh} [J. Reine Angew. Math. 375/376, 259--306 (1987; Zbl 0619.12008)] has shown that the set of realizable classes \({\mathcal R}({\mathfrak M})\) is actually a subgroup of \(\text{Cl}({\mathfrak M})\). The paper under review proves the same result for \(\Gamma\simeq A_4\), provided that the class number of \(k\) is odd and \(\zeta_3\not\in k\). The latter condition guarantees the Eichler condition for \(k[A_4]\). The paper is built on Fröhlich's Hom description of \(\text{Cl}({\mathfrak M})\), his resolvents, and on earlier work of the authors [J. Théor. Nombres Bordx. 14, 241--248 (2002; Zbl 1026.11082)] and of \textit{B. Sodaïgui} [J. Number Theory 28, 189--204 (1988; Zbl 0641.12005)].
Fröhlich's Hom-description of class group
scientific article; zbMATH DE number 1902882