Hi. Peter Roquette sent me an email asking for an example of a quadratic space in characteristic 2 having certain features. I have no idea on this, but maybe someone reading this does.

He would like an example of a field $K$ of characteristic 2 with the following two properties:

(1) The quaternion algebras over $K$ form a group (within the Brauer group of $K$).

(2) There exists an anisotropic quadratic form of dimension > 4 over $K$ that is "completely regular in the sense of Arf". That means: there is a quadratic space $(V,Q)$ over $K$ such that

(a) $\dim(V) > 4$

(b) if $Q(v) = 0$ then $v = 0$

(c) for the associated bilinear form $B(v,w) = Q(v+w) - Q(v) - Q(w)$, if $B(v,w) = 0$ for all $w$ in $V$ then $v = 0$.

Here's the background. Roquette has written a paper with Falko Lorenz on the historical development of the Arf invariant, and they include in the paper a counterexample to the method of proof of a theorem of Arf. (The paper is at Roquette's homepage at http://www.rzuser.uni-heidelberg.de/~ci3/arf.pdf and it has also appeared in "Mathematische Semesterberichte" vol. 57 (2010) pp. 73--102.) Their counterexample to Arf's method of proof is not actually a counterexample to his main theorem. In order to find a counterexample to the main theorem itself they want a quadratic space with the properties above (in characteristic 2). Roquette has asked around but nobody has yet been able to give him an example.