Discriminants and module invariants over a Dedekind domain

Extract

In an earlier paper (cf. [1]) I had given a generalization of the concept of an absolute discriminant to arbitrary finite number fields K as base fields. In a second paper (cf. [2] 2. 3, see also [3] 1. 3) it was shown that the discriminant δ(Λ/K) of a finite extension Λ of K determines the structure of the ring ς of algebraic integers in Λ qua module over the ring ο of algebraic integers in K. The purpose of the present note is to establish a corresponding result for an arbitrary Dedekind domain ο, and finite separable extensions Λ of its quotient field K. The general theory of discriminants and module invariants developed in [1] and [2] for algebraic integers applies in principle to arbitrary Dedekind domains, as already pointed out in the earlier papers. It is usually evident what further hypotheses—if any—have to be imposed to ensure the validity of any particular theorem. For the quoted result of [2] this is, however, not at all clear. In fact the proof involves the proposition:

I. If an element of K is a square everywhere locally then it is a square in K.