For a real quadratic field \(K=\mathbb {Q} ( \sqrt{D} )\) , let \(K_{\infty }\) denote the cyclotomic \(\mathbb {Z}_{p}\) -extension of K. Greenberg conjectured that the corresponding Iwasawa module \(X_{\infty }\) is finite. Building on the work of Mouhib and Movahhedi, we provide new examples of real quadratic fields for which the conjecture holds, when \(X_{\infty }\) is cyclic and the prime is \(p=2\) . Furthermore, we find a fundamental system of units for certain biquadratic fields of the form \(\mathbb {Q} ( \sqrt{2}, \sqrt{D} )\) and show how to use it to calculate the order of \(X_{\infty }\) .
