On the decomposition of x d + a e x e + ⋯ + a 1 x + a 0

Javier Gomez-Calderon

International Journal of Mathematics and Mathematical Sciences, 2000, vol. 23, 1-5

Abstract:

Let K denote a field. A polynomial f ( x ) ∈ K [ x ] is said to be decomposable over K if f ( x ) = g ( h ( x ) ) for some polynomials g ( x ) and h ( x ) ∈ K [ x ] with 1 < deg ( h ) < deg ( f ) . Otherwise f ( x ) is called indecomposable. If f ( x ) = g ( x m ) with m > 1 , then f ( x ) is said to be trivially decomposable. In this paper, we show that x d + a x + b is indecomposable and that if e denotes the largest proper divisor of d , then x d + a d − e − 1 x d − e − 1 + ⋯ + a 1 x + a 0 is either indecomposable or trivially decomposable. We also show that if g d ( x , a ) denotes the Dickson polynomial of degree d and parameter a and g d ( x , a ) = f ( h ( x ) ) , then f ( x ) = g t ( x − c , a ) and h ( x ) = g e ( x , a ) + c .

Date: 2000
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jijmms:283862

DOI: 10.1155/S0161171200002830

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