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Polynomial and complete residue system

Source: Iran TST 2012-Third exam-2nd day-P4

May 16, 2012
algebrapolynomialmodular arithmeticinductionVietacalculusIran

Problem Statement

Suppose pp is an odd prime number. We call the polynomial f(x)=∑j=0najxjf(x)=\sum_{j=0}^n a_jx^j with integer coefficients ii-remainder if ∑p−1∣j,j>0aj≡i(modp) \sum_{p-1|j,j>0}a_{j}\equiv i\pmod{p}. Prove that the set {f(0),f(1),...,f(p−1)}\{f(0),f(1),...,f(p-1)\} is a complete residue system modulo pp if and only if polynomials f(x),(f(x))2,...,(f(x))p−2f(x), (f(x))^2,...,(f(x))^{p-2} are 00-remainder and the polynomial (f(x))p−1(f(x))^{p-1} is 11-remainder.
Proposed by Yahya Motevassel
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