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.

Chowla (1962)、McEliece (1974)、Evans (1977, 1981) 和 Aoki (1997, 2004, 2012) 研究了高斯和，其中一些正整数幂属于有理数领域。这样的高斯和被称为纯。特别是，Aoki (2004) 根据 Dirichlet 字符对所涉及的乘法字符的阶数求模，给出了高斯和为纯的充分必要条件。在本文中，我们研究具有奇扩展度f 的纯高斯和，并将它们分类为$F=5,7,9,11,13,17,19,23$基于青木定理。此外，鉴于偏斜哈达玛差分集的应用，我们表征了纯高斯和的特殊子类。在表征的基础上，我们从有限域的分圆类中给出了斜哈达玛差分集的新构造。