Anisotropy and the integral closure

Let K be a number field and let A be an order in K. The trace map from K to Q induces a non-degenerate symmetric bilinear form 〈,〉:B×B→Q/Z where B is a certain finite abelian group of size Δ(A). In this article we discuss how one can obtain information about OK by purely looking at this symmetric bilinear form. The concepts of anisotropy and quasi-anisotropy, as defined in another article by the author, turn out to be very useful. We will for example show that under certain assumptions one can obtain OK directly from 〈,〉. In this article we will work in a more general setting than we have discussed above. We consider orders over Dedekind domains.