Anisotropic modules over artinian principal ideal rings

Abstract

Let V be a finite-dimensional vector space over a field k and let W be a 1-dimensional k-vector space. Let < , >: V x V \to W be a symmetric bilinear form. Then < , > is called anisotropic if for all nonzero v \in V we have \neq 0. Motivated by a problem in algebraic number theory, we come up with a generalization of the concept of anisotropy to symmetric bilinear forms on finitely generated modules over artinian principal ideal rings. We will give many equivalent definitions of this concept of anisotropy. One of the definitions shows that one can check if a form is anisotropic by checking if certain forms on vector spaces are anisotropic. We will also discuss the concept of quasi-anisotropy of a symmetric bilinear form, which has no useful vector space analogue. Finally we will discuss the radical root of a symmetric bilinear form, which doesn't have a useful vector space analogue either. All three concepts have applications in algebraic number theory.