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 $p_1\equiv p_2 \equiv 1 \pmod{4}$.