The root number class and Chinburg's second invariant
Let \(N/K\) be a finite normal extension of number fields, \(G=\text{Gal}(N/K)\), \(D=\mathbb{Z}[G]\). The ring of integers \({\mathcal O}_ N\) defines a class \(\Omega_ a(N/K)=({\mathcal O}_ N)-[K: \mathbb{Q}](D)\) in the class-group \(C\ell(D)\) (= the finite torsion subgroup of \(K_ 0(D))\). In Invent. Math. 63, 41-79 (1981; Zbl 0469.12003), \textit{M. J. Taylor} proved Fröhlich's conjecture, that \(\Omega_ a(N/K)\) is equal to the root number class \(W_{N/K}\) defined by means of the Artin root numbers of the irreducible symplectic representations of \(G\). Moreover, in Ann. Math., II. Ser. 121, 351-376 (1985; Zbl 0567.12010), \textit{T. Chinburg} defined Galois invariants \(\Omega(N/K,i)\), \(i=1,2,3\) in \(C\ell(D)\) and proved that \(\Omega(N/K,2)=\Omega_ a(N/K)\) for tamely ramified \(N/K\). He also conjectured that \(\Omega(N/K,2)=W_{N/K}\) for all \(N/K\). In the present paper, the author proves Chinburg's conjecture in the case where \(K=\mathbb{Q}\), \(G\simeq\mathbb{H}_ 8\) and 2 is not totally ramified in \(N\).
scientific article; zbMATH DE number 130472
scientific article