Regular Article
A Weil-Bound Free Proof of Schur's Conjecture

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Abstract

Letfbe a polynomial with coefficients in the ring OKof integers of a number field. Suppose thatfinduces a permutation on the residue fields OK/p for infinitely many nonzero prime ideals p of OK. Then Schur's conjecture, namely thatfis a composition of linear and Dickson polynomials, has been proved by M. Fried. All the present versions of the proof use Weil's 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.

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Communicated by Stephen, D. Cohen

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The author thanks the Deutsche Forschungsgemeinschaft (DFG) for its support in form of a postdoctoral fellowship.

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