Elements of order 4 of the Hilbert kernel in quadratic number fields (Q2717602)

The author considers the \(K_2\)-group \(K_2 O_F\), concentrating on its 2-Sylow subgroup for a quadratic number field \(F\) and its integer ring \(O_F\). Using the method of M. Kolster in 1986 and P. E. Conner and J. Hurrelbrink in 1989, he gets again the results of H. Qin for the 2-Sylow subgroups obtained in 1993-1995 [Acta Arith. 69, 153-169 (1995; Zbl 0826.11055), ibid. 72, 323-333 (1995; Zbl 0834.11050) and Sci. China, Ser. A 23, 1254-1263 (in Chinese) (1993)], and expresses elements of order 4 of \(K_2 O_F\) in a simpler form. NEWLINENEWLINENEWLINEThen he obtains relations, e.g. \(r_4(K_2 O_F) = r_4(\Re _F) +1\) in some cases, for the 4-rank of \(K_2 O_F\) and that of the Hilbert kernel \(\Re _2 F\), finding some fields \(F\) with elements of order 8 in \(K_2 O_F\). And for \(F=\mathbb{Q}(\sqrt{p_1 p_2})\) with \(p_1 \equiv p_2 \equiv 5 \pmod 8\), he proves that \(K_2 O_F \cong \mathbb{Z}/(2) \oplus \mathbb{Z}/(2) \oplus \mathbb{Z}/(4) \) iff \(16\mid h(F)\) the class number of \(F\).