International Journal of Mathematics and Mathematical Sciences
Volume 20 (1997), Issue 4, Pages 799-802
doi:10.1155/S0161171297001087
The radical factors of f(x)−f(y) over finite fields
Department of Mathematics, New Kensington Campus, The Pennsylvania State University, New Kensington 15068, PA, USA
Received 6 October 1995; Revised 23 April 1996
Copyright © 1997 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 F denote the finite field of order q For f(x) in F[x], let f*(x,y) denote the substitution polynomial f(x)−f(y). The polynomial f*(x,y) has frequently been used in questions on the values set of f(x) In this paper we consider the irreducible factors of f*(x,y) that are “solvable by
radicals” We show that if R(x,y) denotes the product of all the irreducible factors of f*(x,y) that are
solvable by radicals, then R(x,y)=g(x)−g(y) and f(x)=G(g(x)) for some polynomials g(x) and G(x) in F[x].