Homogeneous spaces, algebraic K-theory and cohomological dimension of fields

Let q be a non-negative integer. We prove that a perfect field K has cohomological dimension at most q C 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.