Note on the quadratic Gauss sums - George Danas (International Journal of Mathematics and Mathematical Sciences)
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)}.