The 2-Sylow subgroup of \(K_2O_F\) for certain quadratic number fields (Q829701)

Let \(F\) be the quadratic number field \(\mathbb{Q}(\sqrt{d})\) where \(d\) is squarefree. Let \(E=F(\sqrt{d/2})\) when \(d\) is even. Suppose that all odd prime divisors of \(d\) are congruent to \(1\) modulo \(8\). The author uses results of H. Qin on the \(4\)-rank of \(K_2(\mathcal{O}_F)\) to determine the relationship between the \(4\)-rank of \(K_2(\mathcal{O}_F)\) and the \(4\)-rank of the ideal class group of \(F\) or of \(E\) under certain additional conditions on the odd prime divisors of \(d\) and the norm of the fundamental unit. The author then gives a simple unified proof of two conjectures of \textit{P. E. Conner} and \textit{J. Hurrelbrink} [Acta Arith. 73, No. 1, 59--65 (1995; Zbl 0844.11072)] which give necessary and sufficient conditions under which \(K_2(\mathcal{O}_F)\) is elementary abelian. These conjectures have already been proved by \textit{A. Vazzana} [Acta Arith. 81, No. 3, 253--264 (1997; Zbl 0905.11051)] in a less concise manner using different methods.