Show that the quaternion division ring contains an infinite number of elements $ u $ satisfying $ u^2=-1 $
I was trying to solve the above exercise on Page 133, Basic Algebra, Jacobson. Maybe it is convenient to consider $$ \mathbb H=\left\{ \left[\begin{matrix} \alpha & \beta\\-\bar{\beta} & \bar{\alpha} \end{matrix} \right]: \alpha, \beta\in\mathbb C \right\}. $$
Of course,
$$ i= \begin{bmatrix} \sqrt{-1} & 0\\ 0 & -\sqrt{-1} \end{bmatrix},\quad j=\begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix},\quad k=\begin{bmatrix}
0 & \sqrt{-1}\\ \sqrt{-1} & 0 \end{bmatrix} $$
satisfy $ u^2=-1 $. But how to show that there are infinitely many elements in $ \mathbb H $ satisfying the identity?
Best Answer
Hint: Forget about the matrices for a moment. What is $(ai+bj)^2$?