On the torsion in \(K_2\) of a field (Q946371)

Let \(F\) be either a number field or a function field over an algebraically closed field \(k\), \(K_2(F)\) the Milnor \(K_2\)-group of \(F\), and \(G_n(F) = \{\{a, \Phi_n(a)\} \in K_2(F) \; | \; a, \Phi_n(a) \in F^*\}\) where \(\Phi_n(a)\) is \(n\)-th cyclotomic polynomial. Assume that \(n \neq 1,4,8,12\) is a positive integer having a square factor and \(\mathrm{char}\,k \nmid n\) in the function field case. The authors prove that \(G_n(F)\) is not a subgroup of \(K_2(F)\). The proofs use the Faltings theorem in the number field case and the results of Manin, Grauert, Samuel and Li on Mordell conjecture in the function field case. In contrast, it is known that if \(n = 1,2,3,4\) or 6, and \(F \neq \mathbb{F}_2\), then \(G_n(G)\) is a subgroup of \(K_2(F)\).