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Some remarks on the \(u\)-invariant. (Quelques remarques sur le \(u\)-invariant.) (French) Zbl 0712.12002

The \(u\)-invariant of a field \(F\) is the supremum of dimensions of anisotropic quadratic forms over \(F\). In the beginning the author supplies some evidence supporting the following conjecture: If \(F\) is a nonreal finitely generated extension of the rationals or of a real closed field, then \(u(F)=2^{\operatorname{cd}(F)}\), where \(\operatorname{cd}(F)\) is the cohomological dimension of \(F\).
For the remaining part of the paper the author says: “Je me propose ici d’estimer \(u(F)\) en termes du groupe de Brauer de \(F\); l’énoncé précis est essentiellement une reformulation de lemmes utilisés par Merkurjev, mais j’espère qu’il présentera malgré tout un intérêt.” More specifically, let \(\operatorname{Quat} F\) be the subgroup of the Brauer group of \(F\) generated by the classes of quaternion algebras over \(F\) and let \(\lambda(F)\) be the smallest integer \(\lambda\) such that every element in \(\operatorname{Quat} F\) can be written as the sum of \(\lambda\) classes of quaternion algebras over \(F\).
Then \(u(F)\ge 2\cdot (\lambda (F)+1)\) for any field \(F\), and if \(I^3 F = 0\) and \(u(F)>1\), then \(u(F)=2\cdot (\lambda (F)+1)\). Here \(I^3 F\) is the third power of the fundamental ideal \(I F\) of the Witt ring \(WF\). Proofs are based on the results of a manuscript of A. S. Merkurjev [Simple algebras over function fields of quadrics (1989)].

MSC:

11E81 Algebraic theory of quadratic forms; Witt groups and rings
12G05 Galois cohomology
11E88 Quadratic spaces; Clifford algebras
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References:

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