Permutation Polynomials of Finite Fields

A permutation polynomial of a finite field K is one for which the associated polynomial function is a one-to-one map from K onto itself. The purpose of this paper is to explain and demonstrate how algebraic geometry can be employed in determining permutation polynomials of finite fields of sufficiently large order. In Chapter II, definitions and well-known results from elementary algebraic geometry, which are used throughout the paper, are stated. As noted earlier, we shall deal only with finite fields whose order is sufficiently large. In Chapter III, results are obtained which are used later to establish criteria for the size of the finite fields to be considered. Also, by elementary means, the exact number of rational points on an absolutely irreducible projective quadratic curve is obtained. In Chapter IV, results given by Chevalley [1], which are used in Chapter III, are displayed, Chapters V through IX are devoted to determining the primary permutation polynomials of degrees four and five of finite fields of sufficiently large order, and the essential results of chapters VI through IX are compiled in Chapter X. The results for the cases which are considered in these chapters were known previously and can be found in Dickson's Linear Groups [2].