Tensor Products of Division Rings and Finite Generation of Subfields

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Let D be a division algebra over a field k. It is shown that if $D{ \otimes _k}{D^0}$ is Noetherian, then every commutative subfield of D containing k is finitely generated over k. This theorem applies to ${D_n}$, the quotient division algebra of the nth Weyl algebra, and also to a number of other standard examples of nonalgebraic division algebras.