The Hermitian structure of rings of integers in odd degree abelian extensions (Q1185821)

A general result is proved on the equivariant genus of Hermitian forms over a group ring \({\mathcal O}_ K[G]\), where \({\mathcal O}_ K\) is the ring of integers in a number field \(K\) and \(G\) is an abelian group of odd order. This result is applied to the case where \(G\) is the Galois group of a tamely ramified extension \(E/K\) and the form is the one obtained by restricting the bilinear form \(t_{E/K}\) to the ring \({\mathcal O}_ E\) of integers in \(E\). More precisely let \(A_{E/K}\) be the unique fractional ideal in \(E\) whose square is the inverse different of the extension \(E/K\); then a locally free ideal \(M_{E/K}\) in \({\mathcal O}_ K[G]\) is constructed such that \(M_{E/K}A_{E/K}={\mathcal O}_ E\). Some further results involving \(M_{E/K}\) are given.