A theorem on permutations in a finite field

Abstract

Let Fq denote the finite field of order q, where q = p7 is odd. Put 41 (a) = + 1, -1 or 0 according as a is a nonzero square, a non-square or 0 in Fq. Then we have (1) +'(a) = am, where q = 2m + 1. A polynomial f(x) with coefficients in Fq is called a permutation polynomial if the numbers f(a), a C Fq, are distinct. For references see [1, Chapter 18; 2, Chapter 5]. The following theorem answers a question raised by W. A. Pierce in a letter to the writer. THEOREM. Let f(x) be a permutation polynomial such that (2) f(?) =0, f(l) = 1 and (3) 41(f(a) -f(b)) = 41(a b) for all a, bE:F,. Then we have (4) f(x) = for some j in the range 0 _?j <n. PROOF. For fixed cE Fq put (5) y = f(c + x)-f(c). It follows from the hypothesis that when x runs through the nonzero squares of Fq the same is true of y; a like result holds for the nonsquares. Thus, if u is an indeterminate, we have TI {u-f(c+x)}= HI {u-f(c)-y}. + (x)=l + (~y)=l