Note on the quadratic Gauss sums

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)}.