$2$-class groups of cyclotomic towers of imaginary biquadratic fields and applications

Let $d$ be a positive square-free integer. In this paper we shall investigate the structure of the $2$-class group of the cyclotomic ${\mathbb{Z}}_2$-extension of the imaginary biquadratic number field $\mathbb{Q}(\sqrt{d},\sqrt{-1})$. Furthermore, we deduce the structure of the $2$-class group of cyclotomic ${\mathbb{Z}}_2$-extension of $\mathbb{Q}(\sqrt{-d})$.