The unit group and the 2-class number of some fields of the form $\mathbb{Q}(\sqrt{2}, \sqrt{pq}, \sqrt{ps})$ and $\mathbb{Q}(\sqrt{2}, \sqrt{pq}, \sqrt{ps}, \sqrt{-\ell})$
Fuente:
arXiv
Enregistré dans:
| Auteur principal: | |
|---|---|
| Format: | Preprint |
| Publié: |
2025
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866908360080621568 |
|---|---|
| author | Hamam, Moha Ben Taleb El |
| author_facet | Hamam, Moha Ben Taleb El |
| contents | 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 $p\equiv-s\equiv 5\pmod 8, q\equiv7\pmod 8 ~~ \text{and} ~~ \left(\frac{ p}{ q}\right)=\left(\frac{ p}{ s}\right)=\left(\frac{s}{ q}\right)=-1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_07092 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The unit group and the 2-class number of some fields of the form $\mathbb{Q}(\sqrt{2}, \sqrt{pq}, \sqrt{ps})$ and $\mathbb{Q}(\sqrt{2}, \sqrt{pq}, \sqrt{ps}, \sqrt{-\ell})$ Hamam, Moha Ben Taleb El Number Theory 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 $p\equiv-s\equiv 5\pmod 8, q\equiv7\pmod 8 ~~ \text{and} ~~ \left(\frac{ p}{ q}\right)=\left(\frac{ p}{ s}\right)=\left(\frac{s}{ q}\right)=-1$. |
| title | The unit group and the 2-class number of some fields of the form $\mathbb{Q}(\sqrt{2}, \sqrt{pq}, \sqrt{ps})$ and $\mathbb{Q}(\sqrt{2}, \sqrt{pq}, \sqrt{ps}, \sqrt{-\ell})$ |
| topic | Number Theory |
| url | https://arxiv.org/abs/2505.07092 |