International Journal of Mathematics and Mathematical Sciences
Volume 23 (2000), Issue 11, Pages 777-781
doi:10.1155/S0161171200002830

On the decomposition of xd+aexe+⋯+a1x+a0

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(xm) with m>1, then f(x) is said to be trivially decomposable. In this paper, we show that xd+ax+b is indecomposable and that if e denotes the largest proper divisor of d, then xd+ad−e−1xd−e−1+⋯+a1x+a0 is either indecomposable or trivially decomposable. We also show that if gd(x,a) denotes the Dickson polynomial of degree d and parameter a and gd(x,a)=f(h(x)), then f(x)=gt(x−c,a) and h(x)=ge(x,a)+c.