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