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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Gauss sums over finite fields and roots of unity


Author: Robert J. Lemke Oliver
Journal: Proc. Amer. Math. Soc. 139 (2011), 1273-1276
MSC (2010): Primary 11T24
Published electronically: September 30, 2010
MathSciNet review: 2748420
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Abstract: Let $ \chi$ be a non-trivial character of $ \mathbb{F}_{q}^\times$, and let $ g(\chi)$ be its associated Gauss sum. It is well known that $ g(\chi)=\varepsilon(\chi)\sqrt{q}$, where $ \vert\varepsilon(\chi)\vert=1$. Using the $ p$-adic gamma function, we give a new proof of a result of Evans which gives necessary and sufficient conditions for $ \varepsilon(\chi)$ to be a root of unity.


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Additional Information

Robert J. Lemke Oliver
Affiliation: Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706
Address at time of publication: Department of Mathematics and Computer Science, Emory University, Atlanta, Georgia 30322
Email: lemkeoliver@gmail.com

DOI: http://dx.doi.org/10.1090/S0002-9939-2010-10662-7
PII: S 0002-9939(2010)10662-7
Keywords: Gauss sums, Gross-Koblitz
Received by editor(s): April 22, 2010
Published electronically: September 30, 2010
Communicated by: Matthew A. Papanikolas
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.