Steinitz classes of Galois extensions with Galois group having nontrivial center
Let \(K/k\) be an extension of number fields, let \(\mathcal O_K\) and \(\mathcal O_k\) be the rings of integers of \(K\) and \(k\) respectively. Then there exists an ideal \(I\) of \(\mathcal O_k\) and an isomorphism \(\mathcal O_K\cong \mathcal O_k^{[K:k]-1}\oplus I\) of \(\mathcal O_k\)-modules. The class \(\mathrm{cl}_k(I)\) of \(I\) in the ideal class group \(\mathrm{Cl}(k)\) of \(k\) is uniquely determined by \(K/k\) and is called its Steinitz class. Given a finite group \(\Gamma\), the set \(R_m(k,\Gamma)\) is defined as the subset of \(\mathrm{Cl}(k)\) of all the Steinitz classes of some tame Galois extensions of \(k\) with Galois group isomorphic to \(\Gamma\). It has been conjectured that \(R_m(k,\Gamma)\) is always a subgroup of \(\mathrm{Cl}(k)\). The conjecture is known to hold when \(\Gamma\) is abelian, as consequence of the main result of \textit{L. R. McCulloh} [J. Reine Angew. Math. 375/376, 259--306 (1987; Zbl 0619.12008)], and for several non-abelian groups. The main difficulty when \(\Gamma\) is non-abelian comes from the fact that one needs to construct some Galois extensions of \(k\) with Galois group isomorphic to \(\Gamma\) and with some given Steinitz class. In the paper under review the author develops a different approach to avoid this kind of problem: he assumes the existence of a tame extension of \(k\) with Galois group \(\Gamma\) and he modifies this extension to obtain some new extensions with the same Galois group, but with different Steinitz classes. In general the Steinitz classes realized in this way do not form a group and should not be expected to be the whole of \(R_m(k,\Gamma)\), but the author gives some conditions under which they coincide with the whole ideal class group of \(k\), which implies that \(R_m(k,\Gamma)=\mathrm{Cl}(k)\). More concretely, the requests are that the center \(Z(\Gamma)\) of \(\Gamma\) is non-trivial and the class number of \(k\) is coprime to an integer \(n_p\), which depends on a prime \(p\) dividing \(|Z(\Gamma)|\), on \(k\) and on \(|\Gamma|\). Several examples are given in the corollaries to the main theorem, showing that this technique allows to prove the conjecture about \(R_m(k,\Gamma)\) in a lot of interesting cases.
scientific article; zbMATH DE number 6124185
scientific article