A Weil-Bound Free Proof of Schurʹs Conjecture

Letfbe a polynomial with coefficients in the ring Kof integers of a number field. Suppose thatfinduces a permutation on the residue fields K/ for infinitely many nonzero prime ideals of K. Then Schurʹs conjecture, namely thatfis a composition of linear and Dickson polynomials, has been proved by M. Fried. All the present versions of the proof use Weilʹs bound on the number of points of absolutely irreducible curves over finite fields in order to get a Galois theoretic translation and to finish the proof by means of finite group theory. This note replaces the use of this deep result by elementary arguments.