论文标题
高斯总和在二次字段中的力量
Powers of Gauss sums in quadratic fields
论文作者
论文摘要
在过去的二十年中，许多研究人员研究了{\ it Index $ 2 $}高斯总和，其中由$ {\ mathbb z}/m {\ mathbb z}/m {\ mathbb z} $的单位组中的特征性$ p $组成的组为索引$ 2 $。 Yang and Xia〜（2010）给出了评估指数$ 2 $高斯总和的完整解决方案。特别是，众所周知，在这种情况下，高斯总和的某些非零积分幂位于二次字段中。另一方面，Chowla〜（1962），McEliece〜（1974），Evans〜（1977，1981）和Aoki〜（1997，2004，2012）研究了{\ it Pure} Gauss总和，某些非零的积分功率在理性的数字领域。在本文中，我们研究高斯总和，其中一些积分功率在二次场中。这类高斯总和是索引$ 2 $高斯总和的概括，以及纯高斯总和到二次字段的扩展。
In the past two decades, many researchers have studied {\it index $2$} Gauss sums, where the group generated by the characteristic $p$ of the underling finite field is of index $2$ in the unit group of ${\mathbb Z}/m{\mathbb Z}$ for the order $m$ of the multiplicative character involved. A complete solution to the problem of evaluating index $2$ Gauss sums was given by Yang and Xia~(2010). In particular, it is known that some nonzero integral powers of the Gauss sums in this case are in quadratic fields. On the other hand, Chowla~(1962), McEliece~(1974), Evans~(1977, 1981) and Aoki~(1997, 2004, 2012) studied {\it pure} Gauss sums, some nonzero integral powers of which are in the field of rational numbers. In this paper, we study Gauss sums, some integral powers of which are in quadratic fields. This class of Gauss sums is a generalization of index $2$ Gauss sums and an extension of pure Gauss sums to quadratic fields.
