The behavior of singular quadratic forms under purely inseparable extensions

Abstract

Let F be a field of characteristic 2 and K a purely inseparable extension of F. When K is modular, we give a complete classification of anisotropic semisingular F-quadratic forms that have over K a maximal Witt index and a defect index at least equal to the half of the dimension of the quasilinear part. The case of totally singular quadratic forms will be also treated. Moreover, without the modularity hypothesis, we give necessary and sufficient conditions under which an anisotropic semisingular F-quadratic form has a given Witt index over K. The quasi-hyperbolicity of semisingular F-quadratic forms over function fields of certain irreducible polynomials will be treated, extending to such forms many results established by the first author in [11].