The behavior of differential, quadratic and bilinear forms under purely inseparable field extensions

Abstract

Abstract. Let F be a field of characteristic p and let E/F be a purely inseparable field extension. We study the group H n+1 p

The behavior of singular quadratic forms under purely inseparable extensions

“…The case of totally singular quadratic forms will be also treated. Our method also allows us to classify the forms φ under the unique hypothesis of maximality of the Witt index over K. This extends a recent result of Sobiech studying the hyperbolicity of nonsingular F -quadratic forms over K [17]. Based on our classifications, we are able to give necessary and sufficient conditions under which an anisotropic semisingular F -quadratic form has a given Witt index over K. We also study the quasi-hyperbolicity of semisingular Fquadratic forms over function fields of certain irreducible polynomials and extend to such forms many results established by the first author in [11].) and the quasilinear part ql(φ) of φ is quasi-hyperbolic over K (Proposition 2.3).…”

“…Our aim in this paper is to study the strict quasi-hyperbolicity of anisotropic semisingular quadratic forms over purely inseparable modular extensions. This is motived by a recent work of Sobiech [17], which gave a complete answer to the hyperbolicity of nonsingular F -quadratic forms over purely inseparable extensions of F (not necessarily modular). Here we restrict ourselves to purely inseparable modular extensions due to the fact that we can't control the quasi-hyperbolicity of totally singular forms over purely inseparable extensions which are not modular.…”

Function field extensions and annihilators of differential forms in characteristic p and bilinear forms in characteristic 2

“…Now to translate the results from section 4 to bilinear forms, we will use a standard procedure which goes back to the case of F having a finite p-basis. A detailed description of the necessary steps can for example be found in [8, p.4] or in [11,Theorem 5.3], so we only state the results in this paper.…”

Splitting of quaternions and octonions over purely inseparable extensions in characteristic 2

“…The isotropy behaviour of quadratic forms over exponent 1 extensions has been studied in [7], including a determination of the Witt kernel for such extensions, i.e., the classifiction of quadratic forms that become hyperbolic over such extensions. Complete results for quartic extensions can be found in [10], and the determination of Witt kernels for arbitrary purely inseparable extensions can be found in [11], [2]. We will not need the full thrust of these results but instead we will focus primarily on quadratic Pfister forms and some explicit examples that have not been exhibited before in the literature in the way we require.…”