Gauss sums over finite fields and roots of unity
Abstract: Let be a non-trivial character of , and let be its associated Gauss sum. It is well known that , where . Using the -adic gamma function, we give a new proof of a result of Evans which gives necessary and sufficient conditions for to be a root of unity.
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Gauss and Jacobi sums.
Canadian Mathematical Society Series of Monographs and Advanced Texts. John Wiley & Sons Inc., New York, 1998.
A Wiley-Interscience Publication. MR 1625181 (99d:11092)
- R. J. Evans.
Generalizations of a theorem of Chowla on Gaussian sums.
Houston J. Math., 3(3):343-349, 1977. MR 0498491 (58:16600)
- B. H. Gross and N. Koblitz.
Gauss sums and the -adic -function.
Ann. of Math. (2), 109(3):569-581, 1979. MR 534763 (80g:12015)
- K. Ireland and M. Rosen.
A classical introduction to modern number theory, volume 84 of Graduate Texts in Mathematics.
Springer-Verlag, New York, second edition, 1990. MR 1070716 (92e:11001)
- J. Yang and W. Zheng.
On a theorem of Chowla.
J. Number Theory, 106(1):50-55, 2004. MR 2049591 (2005b:11197)
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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
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.