A note on the u-invariant of fields (Q1057310)

The u-invariant of a field K of characteristic \(\neq 2\) is defined to be \(\infty\) or the maximum of the dimensions of anisotropic torsion forms over K. The author reproves a well known theorem of \textit{R. Elman} and \textit{T. Y. Lam} [Invent. Math. 21, 125-137 (1973; Zbl 0267.10029)], which states that if quaternion algebras over K form a subgroup in the Brauer group B(K), then the u-invariant of K is equal to 1, 2, 4 or 8. The original proof uses the Arason-Pfister Hauptsatz, which in prevalent opinion is considered to be a strong tool. The main goal of this paper is to give an elementary proof, which does not involve generic methods.