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 mod p 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 − 1 are the eigenvalues of the circulant p × p matrix X with elements the terms of the sequence { χ ( m ) } .