Powers of Gauss sums in quadratic fields
Powers of Gauss sums in quadratic fields
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DOI：
10.1016/j.jnt.2021.08.015
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发表时间：
2020-11
影响因子：
0.7
通讯作者：
K. Momihara
中科院分区：
文献类型：
--
作者：
K. Momihara
在过去的二十年中，许多研究者研究了指数2高斯和，其中由下面有限域的特征p生成的群在Z/mZ的单位群中指数为2，其中所涉及的乘法特征标的阶为m。Yang和Xia（2010）给出了指数2高斯和计算问题的完整解决方案。特别地，已知在这种情况下高斯和的某些非零整数幂在二次域中。另一方面，Chowla（1962），McEliece（1974），Evans（1977，1981）和Aoki（1997，2004，2012）研究了纯高斯和，其中一些非零整数幂在有理数域中。本文研究了二次域中某些整数幂的高斯和。这类高斯和是指数2高斯和的推广，也是纯高斯和在二次域上的推广。
In the past two decades, many researchers have studied 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 Z/m 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 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.