Frontiers of Mathematics in China >

2012 , Vol. 7 >Issue 4: 743 - 764

DOI: https://doi.org/10.1007/s11464-012-0217-2

RESEARCH ARTICLE

Sign or root of unity ambiguities of certain Gauss sums

  • Lingli XIA 1,2 ,
  • Jing YANG , 2
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  • 1. Basic Courses Department, Beijing Union University, Beijing 100101, China
  • 2. Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China

Received date: 02 Dec 2009

Accepted date: 25 Apr 2012

Published date: 01 Aug 2012

Copyright

2014 Higher Education Press and Springer-Verlag Berlin Heidelberg
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Abstract

Gauss sums play an important role in number theory and arithmetic geometry. The main objects of study in this paper are Gauss sums over the finite field with q elements. Recently, the problem of explicit evaluation of Gauss sums in the small index case has been studied in several papers. In the process of the evaluation, it is realized that a sign (or a root of unity) ambiguity unavoidably occurs. These papers determined the ambiguities by the congruences modulo L, where L is certain divisor of the order of Gauss sum. However, such method is unavailable in some situations. This paper presents a new method to determine the sign (root of unity) ambiguities of Gauss sums in the index 2 case and index 4 case, which is not only suitable for all the situations with q being odd, but also comparatively more efficient and uniform than the previous method.

Key words： Gauss sum; Teichmüller characters; Stickelberger’s congruence; Stickelberger’s relation

Cite this article

Lingli XIA , Jing YANG . Sign or root of unity ambiguities of certain Gauss sums[J]. Frontiers of Mathematics in China, 2012 , 7(4) : 743 -764 . DOI: 10.1007/s11464-012-0217-2

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