A generalization of Froehlich's theorem to wildly ramified quaternion extensions of \(\mathbb Q\) (Q2639907)

We study the ``additive-multiplicative'' Galois structure invariant \(\Omega(N/\mathbb Q,2)\) of quaternion extensions \(N/\mathbb Q\). If \(N/\mathbb Q\) is at most tamely ramified, T. Chinburg proved that \(\Omega(N/\mathbb Q,2)\) is the stable isomorphism class \(({\mathfrak O}_ N)\) of the ring of integers \({\mathfrak O}_ N\) of \(N\) in the class group \(Cl(\mathbb Z[H_ 8])=\{\pm 1\}\) of projective modules for \(\mathrm{Gal}(N/\mathbb Q)\cong H_ 8\). In this case, A. Fröhlich proved that \(({\mathfrak O}_ N)\) equals the Artin root number \(W_{N(\mathbb Q)}=\pm 1\) of the unique two-dimensional irreducible representation of \(\mathrm{Gal}(N/\mathbb Q)\). We generalize these results by showing that for all quaternion extensions \(N/\mathbb Q\) having at least two places over the prime 2, one has \(\Omega(N/\mathbb Q,2)=W_{N/\mathbb Q}\).