I have some issues with these definitions and maybe the not enough explicit definition of quaternion algebras. Let $D$ be a division quaternion algebra over $\mathbb{Q}$. The question is the one of the title:
- (1) Are all the elements of $D$ or $D/Z$ regular?
- (2) Are all the elements of $D$ or $D/Z$ elliptic?
- (3) Are all the elements of $D$ or $D/Z$ semisimple?
Thanks for any clue!
Let $D$ be a quaternion division algebra over a field $F$. Then, for $\alpha \in D - F$, the centralizer of $\alpha$ in $D$ is the quadratic extension $E=F[\alpha]$. Note $E$ is not split since $D$ is division, so $\alpha$ is regular, elliptic and semisimple. (Elements in $F$ are not regular.)