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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 \(k\subset k(X)\) and give generators for the annihilator of the ideal. When \(\ell =2\), this was done by Orlov, Vishik and Voevodsky.

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