On the structure of the Iwasawa module for \(\mathbb{Z}_{2}\)-extensions of certain real biquadratic fields

For an infinite family of real biquadratic fields k we give the structure of the Iwasawa module \(X=X(k_{\infty})\) $X=X(k_{\infty })$ of the \(\mathbb{Z}_{2}\) ${\mathbb{Z}}_2$ -extension of k. For these fields, we obtain that \(\lambda=\mu=0 \mbox{ and }\nu=2\) $\lambda =\mu =0\text{and}\nu =2$ . where \(\lambda\) $\lambda$ , \(\mu\) $\mu$ and \(\nu\) $\nu$ are the Iwasawa invariants of the cyclotomic \(\mathbb{Z}_{2}\) ${\mathbb{Z}}_2$ -extension of k