On the roots of polynomials modulo primes

We study the problem of counting the number of roots of an irreducible polynomial f(X) ∈ Z [X] modulo rational primes. We consider the family of polynomials fn(X) = Xn -- X -- 1, which have Galois groups isomorphic to Sn. The approach we take is to attach Galois representations to the counting problem and then to relate these to automorphic forms. In particular, we attempt to attach the representations to holomorphic forms on GL2. We show this only works when n ≤ 5, and we present the solutions to the problem in the n = 4 and 5 cases, following methods due to Serre, Crespo, and Buhler for explicitly constructing Galois representations. The solution to the n = 5 case is novel, requiring Hilbert modular forms. In solving the problem, we produce the first example of an icosahedral Hilbert form that is not the base change of a classical form.