Dimensions of Anisotropic Indefinite Quadratic Forms, I

By a theorem of Elman and Lam, fields over which quadratic forms are classified by the classical invariants dimension, signed discriminant, Clifford invariant and signatures are exactly those fields F for which the third power IF of the fundamental ideal IF in the Witt ring WF is torsion free. We study the possible values of the uinvariant (resp. the Hasse number ũ) of such fields, i.e. the supremum of the dimensions of anisotropic torsion (resp. anisotropic totally indefinite) forms, and we relate these invariants to the symbol length λ, i.e. the smallest integer n such that the class of each product of quaternion algebras in the Brauer group of the field can be represented by the class of a product of ≤ n quaternion algebras. The nonreal case has been treated before by B. Kahn. Here, we treat the real case which turns out to be considerably more involved. 1991 Mathematics Subject Classification: 11E04, 11E10, 11E81, 12D15