Let \alpha_1, \alpha_2, \ldots, \alpha_6 be a fixed labeling of the complex roots of x^6-1. Find the number of permutations \left\{\alpha_{i_1}, \alpha_{i_2}, \ldots, \alpha_{i_6}\right\} of these roots such that if P\left(\alpha_1, \ldots, \alpha_6\right)=0, then P\left(\alpha_{i_1}, \ldots, \alpha_{i_6}\right)=0, where P is any polynomial with rational coefficients.