A class number calculation of the $5^{\text{th}}$ layer of the cyclotomic $\mathbf{Z}_2$-extension of $\mathbf{Q}(\sqrt{5})$

Abstract

For a positive integer $n$, let $K_n$ be the $n$-th layer of the the cyclotomic $\mathbf{Z}_2$-extension of $\mathbf{Q}(\sqrt{5})$, which is the real quadratic field with the minimal discriminant. We prove that the class number of $K_5$ is 1.