Let $D$ be a division ring. Show that if every $a \in D$ is algebraic over the prime subfield of $D$ then $D$ is commutative

Question

Let $D$ be a division ring. Show that if every $a \in D$ is algebraic over the prime subfield of $D$ then $D$ is commutative ($D=Z(D)$).

Answer 1

A counterexample is $\mathbb Q(i,j,k)$ inside the quaternions. (Use that every pure quaternion has a real square.)