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 $I^3F$ of the fundamental ideal $IF$ in the Witt ring $WF$ is torsion free. We study the possible values of the $u$-invariant (resp. the Hasse number $\hn$) 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 $\lambda$, 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 $\leq n$ 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.

2000 Mathematics Subject Classification: 11E04, 11E10, 11E81, 12D15

Keywords and Phrases: quadratic form, indefinite quadratic form, torsion quadratic form, real field, $u$-invariant, Hasse number, symbol length

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