\(\mathbb {Z}_2\)-extension of real quadratic fields with \(\mathbb {Z}/2\mathbb {Z}\) as 2-class group at each layer

Let \(K= \mathbb {Q}(\sqrt{d})\) $K=\mathbb{Q}(\sqrt{d})$ be a real quadratic field with d having three distinct prime factors. We show that the 2-class group of each layer in the \(\mathbb {Z}_2\) ${\mathbb{Z}}_2$ -extension of K is \(\mathbb {Z}/2\mathbb {Z}\) $\mathbb{Z}/2\mathbb{Z}$ under certain elementary assumptions on the prime factors of d. In particular, it validates Greenberg’s conjecture on the vanishing of the Iwasawa \(\lambda \) $\lambda$ -invariant for a new family of infinitely many real quadratic fields.