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Additive Galois module structure and Chinburg’s invariant. (English) Zbl 0737.11033
Let \(N/K\) be a finite Galois extension of number fields with group \(\Gamma\). Let \(Cl(\mathbb{Z}\Gamma)\) be the locally free class group of the integral group ring \(\mathbb{Z}\Gamma\), and let \(D(\mathbb{Z}\Gamma)\subset Cl(\mathbb{Z}\Gamma)\) be its kernel group. T. Chinburg introduced an invariant \(\Omega(N/K,2)\in Cl(\mathbb{Z}\Gamma)\), generalising the class of the ring of integers \({\mathcal O}_ N\), which is locally free if and only if \(N/K\) is tamely ramified. He conjectured that \(\Omega(N/K,2)=W_{N/K}\) where \(W_{N/K}\) is the (generalised) root number class: this is an analogue for wild extensions of Fröhlich’s conjecture (now Taylor’s theorem), that \(({\mathcal O}_ N)=W_{N/K}\in D(\mathbb{Z}\Gamma)\) for tame \(N/K\).
Our main results is \(\Omega(N/K,2)\equiv W_{N/K} (\text{mod} D(\mathbb{Z}\Gamma))\), which supports Chinburg’s conjecture, and is of interest as neither of the invariants lie in \(D(\mathbb{Z}\Gamma)\) in general (unlike the tame case). We use a generalisation of Burns and Fröhlich’s canonical factorisation to non-abelian groups \(\Gamma\), and module- theoretic techniques to relate the Hom-description of the above invariants (modulo \(D(\mathbb{Z}\Gamma))\) to a canonical factorisation related to \({\mathcal O}_ N\).

MSC:
11R33 Integral representations related to algebraic numbers; Galois module structure of rings of integers
11R29 Class numbers, class groups, discriminants
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