Let q be an anisotropic 9-dimensional quadratic form over a field (of characteristic not 2). We show that the projective quadric given by q possesses a Rost projector if and only if q is a Pfister neighbor. The following consequence of this result gives the initial step in Oleg Izhboldin's construction of the filed with the u-invariant 9: if q is not a Pfister neighbor and the Schur index of its even Clifford algebra is at least 4, then q is anisotropic over the function field of any 9-dimensional form non-similar to q.

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Nikita A. Karpenko <karpenk@math.uni-muenster.de>