On the theorem of the primitive element with applications to the representation theory of associative and Lie algebras

Abstract

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 ∈ K. When is F[x, y] = F[αx + βy] for some nonzero elements α, β ∈ F?