On the unit group of some multiquadratic fields

In this paper, we calculate the unit groups and the $2$-class numbers of the fields $\mathbb{K}=\mathbb{Q}(\sqrt{2},\sqrt{p_1},\sqrt{p_2})$ and $\mathbb{L}=\mathbb{Q}(\sqrt{-1},\sqrt{2},\sqrt{p_1},\sqrt{p_2})$, where $p_1$ and $p_2$ are two prime numbers satisfying $p_1\equiv p_2\equiv 1(\mathrm{m}\mathrm{o}\mathrm{d}4)$.