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

We describe of 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 $x,y\in K$. When is $F[x,y]=F[αx+βy]$ for some non-zero elements $α,β\in F$?

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