Alexander Vishik: Fields of u-invariant 2^r+1.

Submission: 2006, Oct 1, revised: 2006, Nov 21

The u-invariant of a field is defined as the maximal dimension of anisotropic quadratic form over it. The natural question is to find the set of possible values of this invariant. From elementary quadratic form theory u can not be 3,5, and 7. In 1991 A.Merkurjev constructed fields with all even values of u, and in 1999 O.Izhboldin constructed a field of u-invariant 9. Still, the question of other values was open. In the current article we provide uniform construction, which gives fields of u-invariant 2n, and 2^r+1, for all r starting from 3. It uses methods different from that of A.Merkurjev and O.Izhboldin. Basic tool here is the so-called "elementary discrete invariant" of a quadric. I would say, that the primary purpose of the article is to demonstrate the importance of this invariant.

2000 Mathematics Subject Classification: 11E04,14C25,14M15

Keywords and Phrases: Quadratic forms, u-invariant, Chow groups, Algebraic cobordisms, Steenrod operations, Landweber-Novikov operations, Grassmannians.

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