I have just asked this question on math.stackexchange, and would like to repost it here:

The norm residue isomorphism theorem establishes that the norm residue map between Milnor K-theory of a field $k$ mod $\ell$ and étale cohomology

$\partial^n: \ K_n^M(k)/\ell \rightarrow H^n_{ét}(k, \mu^{\otimes n}_\ell)$

is an isomorphism. Just how much information about the non-reduced K-theory of a field can be derived from étale cohomology?