On the 2-class number of some real cyclic quartic number fields I

We consider the real cyclic quartic number field \(\mathbb {K}=\mathbb {Q}(\sqrt{n\varepsilon _{0}\sqrt{\ell }})\) $\mathbb{K}=\mathbb{Q}(\sqrt{n{\epsilon }_0\sqrt{\ell }})$ , where \(\ell =2\) $\ell =2$ or \(\ell \equiv 1\pmod 4\) $\ell \equiv 1(mod4)$ is a prime, n is a square-free positive integer relatively prime to \(\ell \) $\ell$ and \(\varepsilon _{0}\) ${\epsilon }_0$ the fundamental unit of its quadratic subfield \(k=\mathbb {Q}(\sqrt{\ell })\) $k=\mathbb{Q}(\sqrt{\ell })$ . In this article, assuming the 2-class group \(\textbf{C}_{\mathbb {K},2}\) ${\mathbf{C}}_{\mathbb{K},2}$ of \(\mathbb {K}\) $\mathbb{K}$ is nontrivial and cyclic, we prove that the order of \(\textbf{C}_{\mathbb {K},2}\) ${\mathbf{C}}_{\mathbb{K},2}$ is exactly 2.