Powers of Gauss sums in quadratic fields

DOI10.1016/J.JNT.2021.08.015arXiv2011.14528WikidataQ114156747 ScholiaQ114156747MaRDI QIDQ6354821

Koji Momihara

Publication date: 29 November 2020

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 $mathbbZ/mmathbbZ$ 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.

Mathematics Subject Classification ID

Quadratic extensions (11R11) Gauss and Kloosterman sums; generalizations (11L05)

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