On Exceptional Polynomials
Kenneth S. Williams · Canadian Mathematical Bulletin · 1968
Let f(x) be a polynomial of degree d ≥ 2 defined over the finite field kq with q = pn elements. Let If f*(x, y) has no irreducible factor over kq which is absolutely irreducible, f is called an exceptional polynomial [1]. Davenport and Lewis have noted that when d is small compared with p, a permutation (substitution) polynomial is necessarily an exceptional polynomial. It is the purpose of this paper to prove the converse; that is, we will show the existence of a constant a(d), depending only on d, such that if f(x.) is an exceptional polynomial over kq, where p ≥ a(d), then f(x) is a permutation polynomial.