On the Theorem of the Primitive Element with Applications to the Representation Theory of Associative and Lie Algebras

Abstract

AbstractWe describe all finite dimensional uniserial representations of a commutative associative (resp. abelian Lie) algebra over a perfect (resp. sufficiently large perfect) field. In the Lie case the size of the field depends on the answer to following question, considered and solved in this paper. LetK/Fbe a finite separable field extension and letx, y∊K. When isF[x,y] =F[αx + βy] for some nonzero elementsα, β∊F?