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.

EDIT (Aug. 16): After Roquette learned about the answer posted below, he has posted on his website http://www.rzuser.uni-heidelberg.de/~ci3/manu.html a paper (number 46) with Lorenz which discusses the solution to his question in the context of Arf's paper. Anyone interested in this topic is encouraged to look at the paper. He wrote to me "Your idea to put my question on the website Math Overflow has worked wonderfully."