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

Let F be a field of characteristic 2, and T ( F ) the set of totally singular F -quadratic forms up to isometry. The u ˆ -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 u ˆ -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 u ˆ -invariant can be given without the use of forms of T ( F ) , this is because most of the known values of the u ˆ -invariant are realized by the forms of T ( F ) (see [12] ). Also, for r , s ≥ 1 integers, we introduce the u r -invariant ( resp. the u ˜ s -invariant) which is the maximal dimension of an anisotropic F -quadratic form having a regular part of dimension 2 r ( 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 u ˆ -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. [ABSTRACT FROM AUTHOR]