On the square root of the inverse different (Q1424593)

Let \(N\) be a finite Galois extension of \(\mathbb{Q}\) with Galois group \(G\) of odd order. Then there is a unique fractional ideal \({\mathcal A}\) of \(N\) with \({\mathcal A}^2={\mathcal D}^{-1}\) where \({\mathcal D}\) is the different of \(N/\mathbb{Q}\). The author considers the structure of \({\mathcal A}\) both as a Galois module and as an integral lattice. Define \(N/\mathbb{Q}\) to be weakly ramified if the second ramification group \(G_2(\wp)\) is trivial for all prime ideals \(\wp\) of \(N\). Suppose that \(N/\mathbb{Q}\) is a weakly ramified \(p\)-extension for an odd prime \(p\). Then it is shown that, in the class group of locally free \(\mathbb{Z} G\)-modules, \(({\mathcal A})^e=1\) where \(e\) is the ramification index of \(p\) in \(N\). It is also shown, using machine computations, that there are weakly ramified extensions \(N/\mathbb{Q}\) of odd degree for which \({\mathcal A}\) is not isometric to the standard lattice \(\mathbb{Z} G\). Among these are examples of both tame and of wild extensions for which \({\mathcal A}\cong\mathbb{Z} G\) as a \(\mathbb{Z} G\)-module.