Exercise 3, § 2

(a) Let K be a commutative field. Show that, for K to be the centre of a quaternion algebra of type $(\alpha, \beta, \gamma)$ over K, it is necessary and sufficient that one of the following cases hold: (1) $\beta \gamma \neq 0$; (2) $\gamma = 0, \beta \neq 0$ and $\beta^2 + 4\alpha \neq 0$; (3) $\beta = 0, \alpha \neq 0$ or $\gamma \neq 0$ and $2 \neq 0$ in K.

(b) Let K be a commutative field such that $2 \neq 0$ in K; show that the quaternion algebra over K of type $(1, \gamma)$ is isomorphic to the matrix algebra $M_2(K)$ (consider the basis of this algebra consisting of the elements $\frac{1}{2}(1+i)$, $\frac{1}{2}(1-i)$, $(j+k/2\gamma)$, $\frac{1}{2}(j-k)$).

(c) Let K be a commutative field such that $2 \neq 0$ in K and let E be the quaternion algebra over K of type $(\alpha, \gamma)$. A quaternion $z \in E$ is called pure if $\overline{z} = -z$ (or also $T(z) = 0$). Show that if $z$ is pure, so is every quaternion $tzt^{-1}$, where $t \in E$ is invertible.

(d) If $u, v$ are any two quaternions, $uv - vu$ is pure. Deduce that if $u, v, w$ are any three quaternions, then

(e) Show that if there exists in $E$ a quaternion $z \neq 0$ such that $N(z) = 0$, there also exists a pure quaternion $z' \neq 0$ such that $N(z') = 0$. (Note, in the notation of formula (33) (no. 5), that, if $\alpha$ is not a square in $K$, the existence of $z \neq 0$ in $E$ such that $N(z) = 0$ is equivalent to the existence of an element $y \in K + Ki$ such that $\gamma = N(y)$.)