Given a number field $K$ which contains at least one primitive root of unity.
Let $L/K$ be a finite extension of $K$. Let $N_{L/K}: K_2(L) \rightarrow K_2(K)$ be the norm map of Milnor or Quillen algebraic $K_2$-groups.
Are there examples of $K$ and $L$ such that $N_{L/K}$ is zero, or examples such that there is a class $\alpha \in K_2(L) $ mapping to zero in $K_2(K)$?