HMMT 二月 2005 · TEAM2 赛 · 第 9 题

HMMT February 2005 — TEAM2 Round — Problem 9

[30] Let x and y be two distinct roots of unity. Prove that x + y is also a root of unity y if and only if is a cube root of unity. x

解析

[30] Let x and y be two distinct roots of unity. Prove that x + y is also a root of unity y if and only if is a cube root of unity. x Solution: This is easiest to see geometrically. The vectors corresponding to x , y , and − x − y sum to 0, so they form a triangle. In order for them all to be roots of unity, they must all have length one, so the triangle must be equilateral. Therefore the angle y 2 π between x and y is ± , that is, is a cube root of unity. 3 x