On the divisibility properties of the orders of \(K_ 2 O_ F\) for certain totally real abelian fields \(F\) (Q1192440)

The author shows that for a given natural number \(n\) there exist infinitely many totally real abelian number fields \(F\), such that \(n\) divides the order of the tame kernel \(K_ 2({\mathfrak o}_ F)\) of \(F\). This generalizes a result of \textit{K. F. Hettling} [Commun. Algebra 17, 501-509 (1989; Zbl 0677.12004)] and is an easy consequence of the following more precise result: Let \(F\) be a totally real subfield of the \(p\)-th cyclotomic field \(\mathbb{Q}(\zeta_ p)\). If \(q\geq 5\) is a prime and \(q^ k\) divides \([F:\mathbb{Q}]\), then \(q^ k\) divides the order of \(K_ 2({\mathfrak o}_ F)\). A similar result holds for the prime 3 provided 9 divides \(p-1\). The proof uses the Birch-Tate Conjecture, which relates the order of \(K_ 2({\mathfrak o}_ F)\) to the value of the zeta-function of \(F\) at \(- 1\).