Appendix on the group \(K_2(F) / \bigcap_{l\geq 1}l K_2(F)\) (Q2783046)

In this appendix to \textit{I. Fesenko}'s paper [St. Petersbg. Math. J. 13, No. 3, 485-501 (2002); translation from Algebra Anal. 13, No. 3, 198-221 (2002; Zbl 1003.11053)], the author constructs an example of a field \(F\) of characteristic different from \(p\) containing a \(p\)th root of unity such that the \(p\)-torsion of \(K_m^{\text{top}}(F)=K_m(F)/\bigcap_{l\geq 1}lK_m(F)\), \(m\geq 2\), is not generated by the \(p\)-torsion of \(F^*\), where \(K_m(F)\) denotes the \(m\)-th Milnor \(K\)-group. The case \(m=2\) is crucial as the general case \(m\geq 2\) can then simply be obtained by adjoining \(m-2\) variables to the field constructed in the case \(m=2\) where it is obtained as a direct limit of function fields of products of certain Severi-Brauer varieties. Main ingredients in the proof include the Merkur'ev-Suslin theorem saying that \(K_2(F)/p^n\) is isomorphic to the \(p^n\)-torsion of the Brauer group of \(F\) provided that \(F\) contain a primitive \(p^n\)-th root of unity, a theorem by \textit{A. A. Suslin} on the torsion in \(K_2(F)\) [K-Theory 1, 5-29 (1987; Zbl 0635.12015)], and a result by \textit{B. Kahn} [Duke Math. J. 69, 137-165 (1993; Zbl 0789.14014)].