The 2-Sylow subgroup of $$K_2O_F$$ for certain quadratic number fields

Abstract

In this paper, we apply Qin’s theorem for the 4-rank of $$K_2O_F$$ to establish the relation between the 4-rank of the ideal class group of $$F=\mathbb {Q}(\sqrt{d})$$ and the 4-rank of $$K_2O_F$$ provided that all odd prime factors of d are congruent to 1 mod 8. As an application, we give a concise and unified proof of two conjectures proposed by Conner and Hurrelbrink (Acta Arith 73:59–65, 1995).