Abstract
In this paper we prove the following theorem:
Let D be a division ring with center the field k, and let k (x 1, …, xn) denote the rational function field in n variables over k. If D contains a maximal subfield which has transcendence degree at least n over k, then D ⊗k k (x1, …, xn) is a simple Noetherian domain of Krull and global dimensions n.
Rather surprisingly, the preceding result can be used to determine the maximum transcendence degrees of the commutative subalgebras of several classically studied division rings. Using the theorem we prove, for example, that in the division ring of quotients of the Weyl algebra,A n, every maximal subfield has transcendence degree at mostn over the center.
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Resco, R. Transcendental division algebras and simple Noetherian rings. Israel J. Math. 32, 236–256 (1979). https://doi.org/10.1007/BF02764919
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DOI: https://doi.org/10.1007/BF02764919