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

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