On quadratic forms of height 3 and degree \(\leq 2\) (Q1291960)

Let \(F\) be a field of characteristic \(\neq 2\), and let \(\phi\) be an anisotropic even-dimensional quadratic form over \(F\). Let \(F_1=F(\phi)\) be the function field of \(\phi\) and let \(\phi_1\) be the anisotropic part of \(\phi\otimes F_1\). \(\phi\) is said to be of height \(1\) if \(\phi_1=0\). It is well known that this holds iff \(\phi\) is similar to some \(d\)-fold Pfister form for some \(d\geq 1\), and \(d\) is called the degree of \(\phi\). Inductively, \(\phi\) is said to be of height \(h\) and degree \(d\) if \(\phi_1\) is of height \(h-1\) and degree \(d\). Forms of height \(2\) have been studied by \textit{M. Knebusch} [Proc. Lond. Math. Soc. (3) 34, 1-31 (1977; Zbl 0359.15013)] who classified those of degree \(1\), \textit{R. W. Fitzgerald} [Trans. Am. Math. Soc. 283, 339-351 (1984; Zbl 0531.10022)] who gave a partial classification of those of degree \(2\), \textit{B. Kahn} [Indag. Math., New Ser. 7, 47-66 (1996; Zbl 0866.11030)] who completed this classification, \textit{J. Hurrelbrink} and \textit{U. Rehmann} [Math. Nachr. 176, 111-127 (1995; Zbl 0876.11017)], and \textit{D. Hoffmann} [C. R. Acad. Sci., Paris, Sér. I 324, 11-14 (1997; Zbl 0879.11016)]. In the present paper, the author attacks the problem of classifying forms of height \(3\) and degree \(\leq 2\). He obtains a list of partial results, including an almost complete classification in the degree \(1\) case. The proofs employ mainly standard results from the algebraic theory of quadratic forms (including some more recent ones by Kahn and by the reviewer), but they are rather technical due to the many case distinctions necessary in the study of forms of height \(3\).