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
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.
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.