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Good old zn

Source: RMO 2004, Grade 12, Problem 2

February 26, 2006
algebrapolynomialfunctionsuperior algebrasuperior algebra solved

Problem Statement

Let f∈Z[X]f \in \mathbb Z[X]. For an n∈Nn \in \mathbb N, n≥2n \geq 2, we define fn:Z/nZ→Z/nZf_n : \mathbb Z / n \mathbb Z \to \mathbb Z / n \mathbb Z through fn(x^)=f(x)^f_n \left( \widehat x \right) = \widehat{f \left( x \right)}, for all x∈Zx \in \mathbb Z. (a) Prove that fnf_n is well defined. (b) Find all polynomials f∈Z[X]f \in \mathbb Z[X] such that for all n∈Nn \in \mathbb N, n≥2n \geq 2, the function fnf_n is surjective. Bogdan Enescu
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