International Journal of Mathematics and Mathematical Sciences
Volume 25 (2001), Issue 3, Pages 167-173

Note on the quadratic Gauss sums

George Danas

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.


Let p be an odd prime and {χ(m)=(m/p)}, m=0,1,...,p1 be a finite arithmetic sequence with elements the values of a Dirichlet character χmodp 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,...,p1are the eigenvalues of the circulant p×p matrix X with elements the terms of the sequence {χ(m)}.