Motivic construction of cohomological invariants

Let G be a group of type E8 of compact type over the field of rational numbers, let K be a field of characteristic 0, and q the 5-fold Pfister form which is the sum of 32 squares. J-P. Serre posed in a letter to M. Rost written on June 23, 1999 the following problem: Is it true that G is split over K if and only if q is hyperbolic over K? In the present article we construct a cohomological invariant of degree 5 for groups of type E8 with trivial Rost invariant over any field k of characteristic 0, and putting the field of rational numbers for k answer positively this question of Serre. Aside from that, we show that a variety which possesses a special correspondence of Rost is a norm variety.

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