HMMT 二月 2008 · 代数 · 第 6 题

英文解析

6. [ 5 ] A root of unity is a complex number that is a solution to z = 1 for some positive integer n .
   Determine the number of roots of unity that are also roots of z + az + b = 0 for some integers a and b .2
   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.
   √
   We find 8 roots of unity: ± 1, ± i , ( ± 1 ± 3 i ).1
   ∞ n − 12
   ∑ ∑ k