Symmetric bilinear forms, quadratic forms and MilnorK-theory in characteristic two

Abstract

is bijective for any field F of characteristic two and for any n>0. Here, K,(F) is the K-group of Milnor defined in [11] and I is the unique maximal ideal of the Witt ring W(F) of non-degenerate symmetric bilinear forms over F. On the other hand, Sah studied quadratic forms in characteristic two in [15]. Let F be a field of characteristic two, Wq(F) the Witt group of non-degenerate quadratic forms over F, and define I'Wq(F) (n>O) by the canonical W(F)-module structure on Wq(F). Then, he proved that Wq(F)/IWq(F) is isomorphic to the group of quadratic cyclic extensions of F and IWq(F)/I2Wq(F) is isomorphic to the subgroup Br(F)2 of the Brauer group Br(F) of F consisting of elements annihilated by 2. In this paper, we prove the above Milnor's conjecture, and describe the precise structures of W(F) and Wq(F) generalizing the result of Sah. For any field F of characteristic p>0, let ~ be the n-th exterior power over F of the absolute differential module n t _g21 Let d(~2~ 1) be the .~F/Z-F/FP" image of the exterior derivation d: f2~ t --~ f2~, and let v(n)v (resp. H"~+1(F)) be the kernel (resp. cokernel) of the homomorphism