On the class numbers of the $n$-th layers in the cyclotomic $\mathbb{Z}_2$-extension of $\mathbb{Q}(\sqrt{5})$

Abstract

Let $K_n$ be the $n$-th layer of the cyclotomic $\mathbb{Z}_2$-extension of $\mathbb{Q}(\sqrt{5})$ and $h(K_n)$ the class number of $K_n$. In this paper, we claim that if an odd prime number $\ell$ satisfies $\ell\equiv 3\pmod{8}$ or $\ell\equiv 5\pmod{8}$, then $\ell$ does not divide $h(K_n)$ for any positive integer $n$.