Notes on the Distribution of Roots Modulo a Prime of a Polynomial

Let f(x) be a monic polynomial in Z[x] with roots α₁, . . ., α_n. We point out the importance of linear relations among 1, α₁, . . . , α_n over rationals with respect to the distribution of local roots of f modulo a prime. We formulate it as a conjectural uniform distribution in some sense, which elucidates data in previous papers.