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

Published:2014-04-03
Printed: Dec 2014
• Leandro Cagliero,
CIEM-CONICET, FAMAF-Universidad Nacional de Córdoba, Córdoba, Argentina.
• Fernando Szechtman,
Department of Mathematics and Statistics, University of Regina, Regina, SK
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## Abstract

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 $F[x,y]=F[\alpha x+\beta y]$ for some non-zero elements $\alpha,\beta\in F$?
 Keywords: uniserial module, Lie algebra, associative algebra, primitive element
 MSC Classifications: 17B10 - Representations, algebraic theory (weights) 13C05 - Structure, classification theorems 12F10 - Separable extensions, Galois theory 12E20 - Finite fields (field-theoretic aspects)

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