Kurosh's problem for division rings: are algebraic division rings locally finite?

Claims Theorem 1.1: there is a countable division ring $D$ of characteristic zero with center $F$ and $x,y\in D$ such that $D=F[x,y]=F(x,y)$, $[D:F]=\infty$, and every element of $D$ is algebraic over $F$ (with no uniform degree bound). So algebraicity over the center does not imply local finiteness. It does NOT give an example in positive characteristic or over a prescribed center, and polynomial degrees grow without bound, as they must by Jacobson's theorem.

Claims Theorem 1.1: there is a countable division ring $D$ of characteristic zero with center $F$ and $x,y\in D$ such that $D=F[x,y]=F(x,y)$, $[D:F]=\infty$, and every element of $D$ is algebraic over $F$ (with no uniform degree bound). So algebraicity over the center does not imply local finiteness. It does NOT give an example in positive characteristic or over a prescribed center, and polynomial degrees grow without bound, as they must by Jacobson's theorem.