International Journal of Mathematics and Mathematical Sciences
Volume 25 (2001), Issue 3, Pages 167-173
doi:10.1155/S016117120100480X
Note on the quadratic Gauss sums
Technological Educational Institution of Thessaloniki, School of Sciences, Department of Mathematics, P.O. Box 14561, Thessaloniki GR-54101, Greece
Received 17 March 2000
Copyright © 2001 George Danas. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
Let p be an odd prime and
{χ(m)=(m/p)}, m=0,1,...,p−1 be a finite arithmetic sequence with elements the values of a
Dirichlet character χ modp which are defined in terms of
the Legendre symbol (m/p), (m,p)=1. We study the relation
between the Gauss and the quadratic Gauss sums. It is shown that
the quadratic Gauss sums G(k;p) are equal to the Gauss sums
G(k,χ) that correspond to this particular Dirichlet
character χ. Finally, using the above result, we prove that
the quadratic Gauss sums G(k;p), k=0,1,...,p−1are
the eigenvalues of the circulant p×p matrix X with
elements the terms of the sequence {χ(m)}.