Abstract
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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