  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.
