HMMT 二月 2005 · TEAM2 赛 · 第 7 题
HMMT February 2005 — TEAM2 Round — Problem 7
题目详情
- [15] Find a real, irreducible quartic polynomial with leading coefficient 1 whose roots are all twelfth roots of unity. k
解析
- [15] Find a real, irreducible quartic polynomial with leading coefficient 1 whose roots are all twelfth roots of unity. Solution: All twelfth roots of unity are roots of 12 6 6 x − 1 = ( x − 1)( x + 1) 3 3 6 = ( x − 1)( x + 1)( x + 1) 2 2 2 4 2 = ( x − 1)( x + x + 1)( x + 1)( x − x + 1)( x + 1)( x − x + 1) , 4 2 so the answer is x − x + 1. k