Principal ideals in mod-$ell$ Milnor $K$-theory

Fix a symbol $underline{a}$ in the mod-$ell$ Milnor $K$-theory of a field $k$, and a norm variety $X$ for $underline{a}$. We show that the ideal generated by $underline{a}$ is the kernel of the $K$-theory map induced by $ksubset k(X)$ and give generators for the annihilator of the ideal. When $ell=2$, this was done by Orlov, Vishik and Voevodsky.