Pure Gauss sums and skew Hadamard difference sets

Abstract

Chowla (1962), McEliece (1974), Evans (1977, 1981) and Aoki (1997, 2004, 2012) studied Gauss sums, some positive integral powers of which are in the field of rational numbers. Such Gauss sums are called pure. In particular, Aoki (2004) gave a necessary and sufficient condition for a Gauss sum to be pure in terms of Dirichlet characters modulo the order of the multiplicative character involved. In this paper, we study pure Gauss sums with odd extension degree f and classify them for f=5,7,9,11,13,17,19,23 based on Aoki's theorem. Furthermore, we characterize a special subclass of pure Gauss sums in view of an application for skew Hadamard difference sets. Based on the characterization, we give a new construction of skew Hadamard difference sets from cyclotomic classes of finite fields.