International Journal of Mathematics and Mathematical Sciences
Volume 23 (2000), Issue 11, Pages 777-781

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

Javier Gomez-Calderon

Department of Mathematics, New Kensington Campus, Pennsylvania State University, New Kensington 15068, PA, USA

Received 9 July 1998; Revised 28 March 1999

Copyright © 2000 Javier Gomez-Calderon. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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+ade1xde1++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(xc,a) and h(x)=ge(x,a)+c.