ON PERMUTATION BINOMIALS OVER FINITE FIELDS

Let F-q be the finite field of characteristic p containing q = p(r) elements and f(x) = ax(n) + x(m), a binomial with coefficients in this field. If some conditions on the greatest common divisor of n - m and q - 1 are satisfied then this polynomial does not permute the elements of the field. We prove in particular that if f(x) = ax(n) + x(m) permutes F-p, where n > m > 0 and a is an element of F-p*, then p- 1 <= (d - 1)d, where d = gcd(n - m, p - 1), and that this bound of p, in terms of d only, is sharp. We show as well how to obtain in certain cases a permutation binomial over a subfield of F-q from a permutation binomial over F-q.