Powers of Gauss sums in quadratic fields

TL;DR

This paper investigates a class of Gauss sums whose integral powers lie in quadratic fields, extending previous work on index 2 and pure Gauss sums to a broader context involving quadratic fields.

Contribution

It introduces a new class of Gauss sums that generalize index 2 and pure Gauss sums, focusing on their properties within quadratic fields.

Findings

Some integral powers of these Gauss sums are in quadratic fields.

The class extends the concept of pure Gauss sums to quadratic fields.

Provides a framework for evaluating these generalized Gauss sums.

Abstract

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.