Anisotropic Modules over Artinian Principal Ideal Rings.

LetVbe a finite-dimensional vector space over a fieldk, and letWbe a 1-dimensionalk-vector space. Let ⟨,⟩:V × V → Wbe a symmetric bilinear form. Then ⟨,⟩ is calledanisotropicif for all nonzerov ∈ Vwe have ⟨ v,v ⟩ ≠ 0. Motivated by a problem in algebraic number theory, we give a generalization of the concept ofanisotropyto 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 ofquasi-anisotropyof a symmetric bilinear form, which has no vector space analogue. Finally, we will discuss theradical rootof a symmetric bilinear form, which does not have a vector space analogue either. All three concepts have applications in algebraic number theory. [ABSTRACT FROM AUTHOR]