Division quaternion algebras over some cyclotomic fields
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866917586552225792 |
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| author | Savin, Diana |
| author_facet | Savin, Diana |
| contents | Let $p_{1}, p_{2}$ be two distinct prime integers, let $n$ be a positive integer, $n$$\geq 3$ and let $ξ_{n} $ be a primitive root of order $n$ of the unity. In this paper we obtain a complete characterization for a quaternion algebra $H\left(p_{1}, p_{2}\right)$ to be a division algebra over the $n$th cyclotomic field $\mathbb{Q}\left(ξ_{n}\right)$, when $n$$\in$$\left\{3,4,6,7,8,9,11,12\right\}$ and also we obtain a characterization for a quaternion algebra $H\left(p_{1}, p_{2}\right)$ to be a division algebra over the $n$th cyclotomic field $\mathbb{Q}\left(ξ_{n}\right)$, when $n$$\in$$\left\{5,10\right\}$. In the 4th section we obtain a complete characterization for a quaternion algebra $H_{\mathbb{Q}\left(ξ_{n}\right)}\left(p_{1}, p_{2}\right)$ to be a division algebra, when $n=l^{k},$ with $l$ a prime integer, $l\equiv 3$ (mod $4$) and $k$ a positive integer. In the last section of this article we obtain a complete characterization for a quaternion algebra $H_{\mathbb{Q}\left(ξ_{l}\right)}\left(p_{1}, p_{2}\right)$ to be a division algebra, when $l$ is a Fermat prime number. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_02497 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Division quaternion algebras over some cyclotomic fields Savin, Diana Number Theory Primary:11S15, 11R52, 11R18, 11R32, 11R04 Let $p_{1}, p_{2}$ be two distinct prime integers, let $n$ be a positive integer, $n$$\geq 3$ and let $ξ_{n} $ be a primitive root of order $n$ of the unity. In this paper we obtain a complete characterization for a quaternion algebra $H\left(p_{1}, p_{2}\right)$ to be a division algebra over the $n$th cyclotomic field $\mathbb{Q}\left(ξ_{n}\right)$, when $n$$\in$$\left\{3,4,6,7,8,9,11,12\right\}$ and also we obtain a characterization for a quaternion algebra $H\left(p_{1}, p_{2}\right)$ to be a division algebra over the $n$th cyclotomic field $\mathbb{Q}\left(ξ_{n}\right)$, when $n$$\in$$\left\{5,10\right\}$. In the 4th section we obtain a complete characterization for a quaternion algebra $H_{\mathbb{Q}\left(ξ_{n}\right)}\left(p_{1}, p_{2}\right)$ to be a division algebra, when $n=l^{k},$ with $l$ a prime integer, $l\equiv 3$ (mod $4$) and $k$ a positive integer. In the last section of this article we obtain a complete characterization for a quaternion algebra $H_{\mathbb{Q}\left(ξ_{l}\right)}\left(p_{1}, p_{2}\right)$ to be a division algebra, when $l$ is a Fermat prime number. |
| title | Division quaternion algebras over some cyclotomic fields |
| topic | Number Theory Primary:11S15, 11R52, 11R18, 11R32, 11R04 |
| url | https://arxiv.org/abs/2303.02497 |