On a conductor discriminant formula of McCulloh (Q1922063)

In a talk in Durham in 1994, in view of generalizing Stickelberger relations to class groups of certain orders, I. R. McCulloh asked if the conductor discriminant formula of class field theory could be extended to all commutative self-dual finite rings \(E\). In this paper, the author answers to McCulloh's question: for any character \(\chi\in \Hom(E^*, \mathbb{C}^*)\), let the conductor \(F_\chi\) be the largest \(E\)-ideal \(I\) for which \(\chi\) factors through \((E/I)^*\), let the norm \({\mathcal N} (F_\chi)\) be the \(\mathbb{Z}\)-ideal generated by the group index \([E: F_\chi]\). Then the conductor product \(\prod_\chi {\mathcal N} (F_\chi)\) is a divisor of the discriminant \(\Delta (T(E)/ \mathbb{Z})\), where \(T(E)\) is a canonical order in a Galois algebra whose Galois group over \(\mathbb{Q}\) is isomorphic to \(E^*\). Moreover, equality holds if and only if \(E\) is a principal ideal ring.