A (down-to-earth) criterion for cohomological dimension

In Serre's book "Galois Cohomology", he proves that a field k has cohomological dimension at most 1 if for all finite separable K/k and all finite Galois L/K, the norm map from L* to K* is surjective; the main ingredient of the proof is an argument from Tate cohomology. I'll talk about how to generalize this to get a criterion for fields of cohomological dimension at most n, involving norm maps on Milnor K-theory.