Quelques invariants de corps de caractéristique 2 liés au [formula omitted]-invariant

Abstract

Let F be a field of characteristic 2, and T(F) the set of totally singular F-quadratic forms up to isometry. The uˆ-invariant of F is the maximal dimension of an anisotropic F-quadratic form. In this paper we introduce new invariants of F related to the uˆ-invariant. The first one, called u˜-invariant, is the maximal dimension of an anisotropic not totally singular F-quadratic form. The motivation of introducing this invariant is to see how the values of the uˆ-invariant can be given without the use of forms of T(F), this is because most of the known values of the uˆ-invariant are realized by the forms of T(F) (see [12]). Also, for r,s≥1 integers, we introduce the ur-invariant (resp. the u˜s-invariant) which is the maximal dimension of an anisotropic F-quadratic form having a regular part of dimension 2r (resp. the maximal dimension of an anisotropic not totally singular F-quadratic form having a quasilinear part of dimension s). We make a comparison between these new invariants by relating them to the u-invariant and the uˆ-invariant, and we give a list of values that they can take. We also discuss the classical question, due to Baeza [3], on the universality of some nonsingular F-quadratic forms when F has finite u-invariant.