In this article, we present a beautiful connection between Hadamard matrices and exponential sums of quadratic symmetric polynomials over Galois fields. This connection appears when the recursive nature of these sequences is analyzed. We calculate the spectrum for the Hadamard matrices that dominate these recurrences. The eigenvalues depend on the Legendre symbol and the quadratic Gauss
sum over finite field extensions. In particular, these formulas allow us to calculate closed formulas for the exponential sums over Galois field of quadratic symmetric polynomials.
Finally, in the particular case of finite extensions of the binary field, we show that the corresponding Hadamard matrix is a permutation away from a classical construction of these matrices.
