On a Conjecture of Carlitz

Abstract

A conjecture of Carlitz on permutation polynomials is as follow: Given an even positive integer n, there is a constant Cn, such that if Fq is a finite field of odd order q with q > Cn, then there are no permutation polynomials of degree n over Fq. The conjecture is a well-known problem in this area. It is easily proved if n is a power of 2. The only other cases in which solutions have been published are n = 6 (Dickson [5]) and n = 10 (Hayes [7]); see Lidl [11], Lausch and Nobauer [9], and Lidl and Niederreiter [10] for remarks on this problem. In this paper, we prove that the Carlitz conjecture is true if n = 12 or n = 14, and give an equivalent version of the conjecture in terms of exceptional polynomials.