HMMT 二月 2008 · 代数 · 第 6 题

HMMT February 2008 — Algebra — Problem 6

[ 5 ] A root of unity is a complex number that is a solution to z = 1 for some positive integer n . 2 Determine the number of roots of unity that are also roots of z + az + b = 0 for some integers a and b . ∞ n − 1 ∑ ∑ k

解析

[ 5 ] A root of unity is a complex number that is a solution to z = 1 for some positive integer n . 2 Determine the number of roots of unity that are also roots of z + az + b = 0 for some integers a and b . Answer: 8 The only real roots of unity are 1 and − 1. If ζ is a complex root of unity that is also a root 2 ¯ ¯ ¯ of the equation z + az + b , then its conjugate ζ must also be a root. In this case, | a | = | ζ + ζ | ≤ | ζ | + | ζ | = 2 2 2 2 2 ¯ 2 and b = ζ ζ = 1. So we only need to check the quadratics z +2 z +1 , z + z +1 , z +1 , z − z +1 , z − 2 z +1. √ 1 We find 8 roots of unity: ± 1, ± i , ( ± 1 ± 3 i ). 2 ∞ n − 1 ∑ ∑ k