Galois module structure for dihedral extensions of degree 8: realizable classes over the group ring
L. McCulloh has posed the problem (and also given the answer in the abelian case already) how to determine the subset \({\mathfrak R}({\mathfrak o}_K[G])\) of the locally free class group \(\text{Cl}({\mathfrak o}_K[G])\), which consists of the classes \([{\mathfrak o}_L]\) arising from the rings \({\mathfrak o}_L\) of integers in the tamely ramified Galois extensions \(L\) of a given number field \(K\) such that \(G_{L/K}\) is isomorphic to a given finite group \(G\). In this paper, which resumes earlier work of the second author [e.g., J. Algebra 213, 549--556 (1999; Zbl 0989.11060) and 223, 367--378 (2000; Zbl 0953.11036)], it is shown that, when \(G= D_4\) and the ray class group of \({\mathfrak o}_K \)with modulus \(4{\mathfrak o}_K\) has odd order, this set \({\mathfrak R}({\mathfrak o}_K[G])\) is the kernel \({\mathfrak a}\) of \(\text{aug}: \text{Cl}({\mathfrak o}_K[G])\to\text{cl}_K\), where `'aug' is induced by the augmentation map \(K[G]\to K\) (by means of Fröhlich's Hom description of \(\text{Cl}({\mathfrak o}_K[G])\)). The hypothesis on \(K\) implies that \(K\) is totally real and that \({\mathfrak a}\) is isomorphic to the product of four copies of the ray class group of \({\mathfrak o}_K\) with modulus \(2{\mathfrak o}_K\). To prove the theorem, the authors construct a biquadratic extension of \(K\) which can be embedded in some \(L'\) with \(G_{L'/K}= D_4\) and, by means of a quadratic twist applied to \(L'\), they obtain an extension \(L\) with a desired Galois module class. In general, \({\mathfrak R}({\mathfrak o}_K[G])\neq 1\). -- An appendix has a correction of proofs given in the above cited papers and in [\textit{B. Sodaïgui}, J. Number Theory 65, No. 1, 87--95 (1997; Zbl 0873.11061)], where a formula of Fröhlich relating certain resolvents was misapplied.
scientific article; zbMATH DE number 2164816
scientific article