  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.
