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IMC 2001 Problem 4

Source: IMC 2001 Day 1 Problem 4

October 30, 2020
roots of unityPolynomials

Problem Statement

p(x)p(x) is a polynomial of degree nn with every coefficient 00 or ±1\pm1, and p(x)p(x) is divisible by (x−1)k (x - 1)^k for some integer k>0 k > 0. qq is a prime such that qln⁡q<kln⁡n+1\frac{q}{\ln q}< \frac{k}{\ln n+1}. Show that the complex qq-th roots of unity must be roots of p(x). p(x).
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