Annihilators of quadratic and bilinear forms over fields of characteristic two

Let F be a field with 2=0, W(F) the Witt ring of symmetric bilinear forms over F and Wq(F) the W(F)-module of quadratic forms over F. Let IF W(F) be the maximal ideal. We compute explicitly in and ImWq(F) the annihilators of n-fold bilinear and quadratic Pfister forms, thereby answering positively, in the case 2=0, certain conjectures stated by Krüskemper in [M. Krüskemper, On annihilators in graded Witt rings and in Milnorʹs K-theory, in: B. Jacob et al. (Eds.), Recent Advances in Real Algebraic Geometry and Quadratic Forms, in: Contemp. Math., vol. 155, 1994, pp. 307–320].