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 × V → W be a symmetric bilinear form. Then ⟨,⟩ is called anisotropic if for all nonzero v ∈ V we have ⟨ v, v ⟩ ≠ 0. Motivated by a problem in algebraic number theory, we give 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 a form is anisotropic if and only 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 vector space analogue. Finally, we will discuss the radical root of a symmetric bilinear form, which does not have a vector space analogue either. All three concepts have applications in algebraic number theory.