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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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