Division quaternion algebras over some cyclotomic fields

Let $p_1,p_2$ be two distinct prime integers, let $n$ be a positive integer, $n$$\ge 3$ and let ${\xi }_n$ be a primitive root of order $n$ of the unity. In this paper we obtain a complete characterization for a quaternion algebra $H(p_1,p_2)$ to be a division algebra over the $n$th cyclotomic field $\mathbb{Q}\left({\xi }_n\right)$, when $n$$\in$$\{3,4,6,7,8,9,11,12\}$ and also we obtain a characterization for a quaternion algebra $H(p_1,p_2)$ to be a division algebra over the $n$th cyclotomic field $\mathbb{Q}\left({\xi }_n\right)$, when $n$$\in$$\{5,10\}$. In the 4th section we obtain a complete characterization for a quaternion algebra $H_{\mathbb{Q}\left({\xi }_n\right)}(p_1,p_2)$ 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 $H_{\mathbb{Q}\left({\xi }_l\right)}(p_1,p_2)$ to be a division algebra, when $l$ is a Fermat prime number.