Elements of finite order in \(K_ 2\) of fields
The author proves two results concerning the Milnor symbol \(\{a,b\}\) in \(K_ 2 (F)\), \(F\) a field. (i) If the characteristic of \(F\) is not 2, then every element of order 4 in \(K_ 2 (F)\) has the form \(\{a, a^ 2 + 1\} \{-1, b\}\), where \(a\), \(a^ 2 + 1\) and \(b\) are all nonzero elements of \(F\). (It is known that the second factor is a typical element of order 2.) (ii) The subset \(\{a, \Phi_{2^ n} (a)\}\) of \(K_ 2 (\mathbb{Q})\), \(\Phi_{2^ n}\) a cyclotomic polynomial, is a subgroup only for \(n = 1,2\).
scientific article; zbMATH DE number 640472