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).

中文翻译：

---

某些二次数字段的$$ K_2O_F $$ K 2 OF的2-Sylow子群

在本文中，我们对\（K_2O_F \）的4秩应用秦定理，以建立\（F = \ mathbb {Q}（\ sqrt {d}）的理想类组的4秩之间的关系\）和\（K_2O_F \）的4位，条件是d的所有奇数素数都与1 mod 8一致。作为应用，我们给出了Conner和Hurrelbrink（Acta Arith）提出的两个猜想的简洁统一的证明。 73：59-65，1995）。