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 $\ell \ge 1$ 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 $p\equiv -s\equiv 5(\mathrm{mod}8),q\equiv 7(\mathrm{mod}8)\text{and}\left(\frac{p}{q}\right)=\left(\frac{p}{s}\right)=\left(\frac{s}{q}\right)=1$ and $\left(\frac{p}{q}\right)=\left(\frac{p}{s}\right),$ or $p\equiv -s\equiv 5(\mathrm{mod}8),q\equiv 7(\mathrm{mod}8)\text{and}\left(\frac{p}{q}\right)=\left(\frac{p}{s}\right)=\left(\frac{s}{q}\right)=-1$.