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 \(X=X(k_{\infty})\) of the \(\mathbb{Z}_{2}\)-extension of k. For these fields, we obtain that \(\lambda=\mu=0 \mbox{ and }\nu=2\). where \(\lambda\), \(\mu\) and \(\nu\) are the Iwasawa invariants of the cyclotomic \(\mathbb{Z}_{2}\)-extension of k