Let q be a non-negative integer. We prove that a perfect field K has
cohomological dimension at most q+1 if, and only if, for any finite extension
L of K and for any homogeneous space Z under a smooth linear connected
algebraic group over L, the q-th Milnor K-theory group of L is spanned
by the images of the norms coming from finite extensions of L over which Z
has a rational point. We also prove a variant of this result for imperfect
fields.Comment: Final accepted versio