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

We 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. Let K/F be a finite separable field extension and let x, y is an element of K. When is F[x, y] = F[alpha x + beta y] for some nonzero elements alpha, beta is an element of F?