Abstract. We study Pfister neighbors and their characterization over fields of characteristic 2, where we include the case of singular forms. We give a somewhat simplified proof of a theorem of Fitzgerald which provides a criterion for when a nonsingular quadratic form q is similar to a Pfister form in terms of the hyperbolicity of this form over the function field of a form ϕ which is dominated by q. From this, we derive an analogue in characteristic 2 of a result by Knebusch saying that, in characteristic = 2, a form is a Pfister neighbor if its anisotropic part over its own function field is defined over the base field. Our result includes certain cases of singular forms, but we also give examples which show that Knebusch's result generally fails in characteristic 2 for singular forms. As an application, we characterize certain forms of height 1 in the sense of Knebusch whose quasi-linear parts are of small dimension. We also develop some of the basics of a theory of totally singular quadratic forms. This is used to give a new interpretation of the notion of the height of a standard splitting tower as introduced by the second author in an earlier paper.

We extend to characteristic 2 a theorem by the first author which states that if ϕ and ψ are anisotropic quadratic forms over a field F such that dim ϕ 2 n < dim ψ for some nonnegative integer n, then ϕ stays anisotropic over the function field F (ψ) of ψ. The case of singular forms is systematically included. We give applications to the characterization of quadratic forms with maximal splitting. We also prove a characteristic 2 version of a theorem by Izhboldin on the isotropy of ϕ over F (ψ) in the case dim ϕ = 2 n + 1 dim ψ.

This article deals with the standard splitting of bilinear forms in characteristic 2. The first part is devoted to the study of bilinear Pfister neighbors (the definition of such a bilinear form is slightly different from the classical definition of a Pfister neighbor quadratic form). In the second part, we introduce the degree invariant for bilinear forms and we prove that for any integer d ≥ 0, the d-th power of the ideal of even dimensional bilinear forms coincides with the set of bilinear forms of degree ≥ d (this is a positive answer to the analogue of the degree conjecture for quadratic forms).In the third part, we classify good bilinear forms of height 2, and we give information on the possible dimensions of bilinear forms of height 2 which are not necessarily good.

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