The unit group and the 2-class number of some fields of the form $\mathbb{Q}(\sqrt{2}, \sqrt{pq}, \sqrt{ps})$ and $\mathbb{Q}(\sqrt{2}, \sqrt{pq}, \sqrt{ps}, \sqrt{-\ell})$

Abstract

Let $\LL^+=\mathbb{Q}(\sqrt{2}, \sqrt{pq}, \sqrt{ps})$ and $\LL=\mathbb{Q}(\sqrt{2}, \sqrt{pq}, \sqrt{ps}, \sqrt{-\ell})$ be two fields, where $q$, $p$ and $s$ three different prime integers and $\ell\geq1$ be a positive odd square-free integer relatively prime to $q$, $p$ and $s$. The purpose of this paper is to show how one can proceed to perform the calculation of unit group of the fields of the form $\LL^+=\mathbb{Q}(\sqrt{2}, \sqrt{pq}, \sqrt{ps})$ and $\LL=\mathbb{Q}(\sqrt{2}, \sqrt{pq}, \sqrt{ps}, \sqrt{-\ell})$. More precisely, we compute the unit group and the $2$-class number of these fields whenever $p\equiv-s\equiv 5\pmod 8, q\equiv7\pmod 8 ~~ \text{and} ~~ \left(\frac{ p}{ q}\right)=\left(\frac{ p}{ s}\right)=\left(\frac{s}{ q}\right)=1$ and $\left(\frac{ p}{ q}\right)=\left(\frac{ p}{ s}\right),$ or $p\equiv-s\equiv 5\pmod 8, q\equiv7\pmod 8 ~~ \text{and} ~~ \left(\frac{ p}{ q}\right)=\left(\frac{ p}{ s}\right)=\left(\frac{s}{ q}\right)=-1$.