A Prime Sensitive Hankel Determinant of Jacobi Symbol Enumerators

We show that the determinant of a Hankel matrix of odd dimension n whose entries are the enumerators of the Jacobi symbols which depend on the row and the column indices vanishes if and only if n is composite. If the dimension is a prime p, then the determinant evaluates to a polynomial of degree p − 1 which is the product of a power of p and the generating polynomial of the partial sums of Legendre symbols. The sign of the determinant is determined by the quadratic character of −1 modulo p. The proof of the evaluation makes use of elementary properties of Legendre symbols, quadratic Gauss sums, and orthogonality of trigonometric functions.