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Let $D$ be a division ring with centre $F$ and denote by $D'$ the derived group (commutator subgroup) of $D^*=D-\{0\}$. It is shown that if each element of $D'$ is algebraic over $F$, then $D$ is algebraic over $F$. It is also proved that each finite separable extension of $F$ in $D$ is of the form $F(c)$ for some element $c$ in the derived group $D'$. Using
these results, it is shown that if each element of the derived group $D'$ is of bounded degree over $F$, then $D$ is finite dimensional over $F$.
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