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On Elman and Lam’s filtration of the \(u\)-invariant. (English) Zbl 0882.11021

Let \(F\) be a field of characteristic different from 2. Let \(\sigma_n\) be the quadratic form over \(F\) which is defined by a sum of \(n\) squares. A quadratic form \(\varphi\) over \(F\) is called torsion if \(\sigma_n\otimes\varphi\) is hyperbolic for some \(n\). The \(u\)-invariant of \(F\) is defined to be the supremum of the dimensions of torsion anisotropic quadratic forms over \(F\). The invariant \(u^{(k)}\) of \(F\) is the supremum of the dimensions of those anisotropic forms \(\varphi\) over \(F\) for which \(\sigma_{2^k}\otimes\varphi\) is hyperbolic. One has \(u^{(k)}\leq u^{(k+1)}\) and \(u=\sup\{u^{(k)}\}\). These invariants \(u^{(k)}\) have been introduced by R. Elman and T.-Y. Lam in [Math. Z. 131, 283-304 (1973; Zbl 0252.10020)], where they also asked whether \(u=u^{(1)}\) holds for all fields.
In this note, the author answers the above question in the negative. His counterexamples are based on Merkurjev’s method to construct fields which have an arbitrarily given even integer \(n\geq 0\) as \(u\)-invariant. He shows that to each integer \(k\geq 1\) and to each sufficiently large even integer \(n\) there exist fields with \(u^{(k)}+ 2=u^{(k+1)}= u= n\), resp. \(u^{(k)}+ 4= u^{(k+1)}= u= n\), resp. \(u^{(k)}+ 4= u^{(k+1)}+ 2= u^{(k+2)}= u= n\).

MSC:

11E81 Algebraic theory of quadratic forms; Witt groups and rings
11E04 Quadratic forms over general fields

Citations:

Zbl 0252.10020
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