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HMMT 二月 2005 · TEAM2 赛 · 第 7 题

HMMT February 2005 — TEAM2 Round — Problem 7

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

英文原题

  1. [15] Find a real, irreducible quartic polynomial with leading coefficient 1 whose rootsare all twelfth roots of unity. k
解析

英文解析

  1. [15] Find a real, irreducible quartic polynomial with leading coefficient 1 whose rootsare 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