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

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].