The Galois structure of the square root of the inverse different (Q749604)

Let K/F be an odd degree Galois extension of number fields with \(G=Gal(K/F)\). It is easy to show that in K there exists a unique ideal A(K/F) whose square is the inverse different of K/F. This ideal is the only which has the property of being self-dual with respect to the bilinear trace form of the extension, thus giving rise to a self-dual Hermitian form over the group ring \({\mathbb{Z}}_ FG\) with coefficients in the ring of integers \({\mathbb{Z}}_ F\) of F. The study of this form was first suggested in [\textit{P. Conner} and \textit{R. Perlis}, A survey of trace forms of algebraic number fields (Singapore, World Scientific 1984; Zbl 0551.10017)]. One possible application of precise results on the square root of the inverse different is the description of the non necessarily self-dual Hermitian form defined by the ring of integers in K. After having studied in detail the special case of absolute Galois extensions of odd prime degree [see the author, J. Algebra 118, 438-446 (1988; Zbl 0663.12015)], it became clear how to extend the results on the module structure to arbitrary odd degree Galois extensions. In this paper we begin by showing that for A(K/F) to be locally free over \({\mathbb{Z}}_ FG\) it is necessary and sufficient that K/F is weakly ramified, i.e. all its second ramification groups are trivial. Next let M be a maximal order in \({\mathbb{Q}}G\) containing \({\mathbb{Z}}G\), we show that if K/F is weakly ramified then \(M\otimes A(K/F)\) is free over M. Even better if K/F is tame, then we show that A(K/F) is actually free over \({\mathbb{Z}}G\). The proofs rely heavily on the techniques developped by A. Fröhlich, M. J. Taylor, Ph. Cassou-Noguès et alia for the study of rings of integers. We only mention that instead of Galois-Gauss we are led to consider generalized Jacobi sums defined in terms of Gauss sums and the second Adams operation on the group of virtual characters of G.