The graded Witt ring of a quasi-pythagorean field (Q1100493)

For a field F of characteristic \(\neq 2\), we denote by \(GWF=\oplus^{\infty}_{n=0}I^ n/I^{n+1}\) the graded Witt ring of classes of nonsingular symmetric bilinear forms, by \(kF=KF/2KF\) the K- ring of F defined by \textit{J. Milnor} [Invent. Math. 9, 318-344 (1970; Zbl 0199.555)], taken modulo two, and by H(F,2) the cohomology ring of the maximal two-extension of F with coefficients in \({\mathbb{Z}}/2{\mathbb{Z}}\). Both kF and H(F,2) carry a standard grading. In 1970, Milnor [loc. cit.] asked whether these three graded rings are isomorphic and conjectured explicit formulae for maps between kF and GWF, and, kF and H(F,2), which he proposed as candidates for such isomorphisms. As of this date, for arbitrary fields, it is not even known whether Milnor's formulae lead to well-defined maps, let alone whether these maps, if well defined, are isomorphisms. However, for large classes of fields, these questions have been settled affirmatively. This is true, for example, for pythagorean fields, i.e., fields F such that for all x,y in F, there is a z in F, with \(z^ 2=x^ 2+y^ 2\), satisfying additional hypotheses [\textit{R. Elman} and \textit{T. Y. Lam}, Am. J. Math. 94, 1155-1194 (1972; Zbl 0259.12101) and \textit{B. Jacob}, J. Algebra 68, 247-267 (1981; Zbl 0457.10007)]. The author calls F quasi-pythagorean if the set \(\{x^ 2+y^ 2|\) x,y in \(F\}\) coincides with the Kaplansky radical of F, i.e. all nonzero a in F such that the binary form \(x^ 2-ay^ 2\) is universal. He then shows that for a quasi-pythagorean formally real field, of finite chain length, as defined by \textit{M. Marshall} [Can. J. Math. 32, 603-627 (1980; Zbl 0433.10009)], the formula given by Milnor [loc. cit.] yields an isomorphism kF\(\cong GWF\). If the assumption on chain length is replaced by assuming that the number of orders of F is finite and the stability index of F is at most 1, he also proves that Milnor's formulae yield an isomorphism kF\(\cong H(F,2)\). As the author notes, this last result is already contained in a paper of \textit{J. K. Arason}, \textit{R. Elman} and \textit{B. Jacob} [Math. Ann. 272, 267-280 (1985; Zbl 0545.10013)]. There are a few less definitive results along these lines.