Algebraically closed noncommutative polynomial rings |
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Authors: | Kirby C Smith |
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Institution: | 1. Department of Mathematics , Texas A &2. M University , College Station, Texas, 77843 |
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Abstract: | Let R be a noncommutative polynomial ring over the division ring K where K has center F. Then R = Kx,σ,D]where σ is a monomorphism of K and D is a σ-derivaton K. R is called dimension finite if (K: Fσ)<∞ and (K: FD)<∞ where Fσ is the subfield of F fixed under σand FD is the subfied of F of D-constants. R is algebraically closed if every nonconstant polynomial in Rfactors completely into linear factors. The algebraically closed dimension finite polynomial rings are determined. s done by reducing the problem to two classes: skew polynomial rings and differential polynomial rings. Examples algebraically closed polynomial rings which are not dimensfinite are given. |
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