After reading the proof of the theorem
“For every central division $F$-algebra $D$ with $D$ $\neq$ $F$, $D$ contains a separable extension $K \supsetneqq F$“,
I have a question: dose there exist a non-commutative division ring $D$ with finite characteristic, which contains an $x\in D$ such that $x^n \not\in Z(D)$ for all positive integers n?
First I tried quaternion algebra over field $F_p(X)$.
Let $i^2= x$, $j^2= y$, $ij=k$, $ji=-k$ where $x, y$ are any non-zero elements in $F_p(X)$, the quaternion algebra $H(F_p(X))$ is a vector space over $F_p(X)$ with a basis $\{1,i ,j, k\}$.
Let $α =a+ib+jc+kd$, $\bar{α}=a-ib-jc-kd$, $α\bar{α}= \bar{α}α= a^2-xb^2-yc^2+xyd^2$, then $H(F_p(X))$ is a division ring if and only if $α\bar{α}= 0$ implies $α= 0$. After trying some examples, I find that the consequent processes are hard. Does there exist some good example that satisfies the property in the question?