On the unit group and the 2-class number of \(\mathbb {Q}(\sqrt{2},\sqrt{p},\sqrt{q})\)

Let \(p\equiv 1\pmod {8}\) $p\equiv 1(mod8)$ and \(q\equiv 7\pmod 8\) $q\equiv 7(mod8)$ be two prime numbers. The purpose of this paper is to compute the unit groups of the fields \(\mathbb {L}=\mathbb {Q}(\sqrt{2}, \sqrt{p}, \sqrt{q})\) $\mathbb{L}=\mathbb{Q}(\sqrt{2},\sqrt{p},\sqrt{q})$ and give their 2-class numbers.