Anisotropy and the integral closure

TL;DR

This paper explores how properties of a symmetric bilinear form derived from a number field's trace map can reveal information about the field's ring of integers, extending to orders over Dedekind domains.

Contribution

It introduces the use of anisotropy and quasi-anisotropy concepts to determine the ring of integers from bilinear form data, generalizing previous results to broader settings.

Findings

The bilinear form <,> encodes information about _K.

Anisotropy properties help recover _K from <,>.

Results extend to orders over Dedekind domains.

Abstract

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 x B \to Q/Z where B is a certain finite abelian group of size \Delta(A). In this article we discuss how one can obtain information about \mathcal{O}_K 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 \mathcal{O}_K directly from <,>. In this article we will work in a more general setting than we have discussed above. We consider orders over Dedekind domains.