Let \( p\equiv 5\pmod 8\) , \( q\equiv 7\pmod 8 \) and \(r\equiv 3 \pmod 8\) be three prime numbers such that \(\left( \frac{p}{q}\right) =\left( \frac{p}{r}\right) =1\) . The purpose of this paper is to show how to compute the unit group of the real triquadratic fields of the form \(\mathbb {L}_1:=\mathbb {Q}(\sqrt{2}, \sqrt{pr}, \sqrt{q})\) and \(\mathbb {K}_1:=\mathbb {Q}(\sqrt{2}, \sqrt{pq}, \sqrt{pr})\) . Furthermore, we investigate the second 2-class group of \(\mathbb {K}_n\) , the nth layer of the cyclotomic \(\mathbb {Z}_2\) -extension of the field \(\mathbb {K}:=\mathbb {Q}(\sqrt{pq}, \sqrt{pr})\) , more precisely, we give an improvement of a part of the main theorem of Chems-Eddin (On the maximal unramified pro-2-extension of \(\mathbb {Z}_2\) -extension of certain real biquadratic fields. arXiv:2409.13574. 2024).
