What is the smallest positive integer $n$ such that all the roots of $z^4 + z^2 + 1 = 0$ are $n^{ ext{th}}$ roots of unity?

Question

What is the smallest positive integer $n$ such that all the roots of $z^4 + z^2 + 1 = 0$ are $n^{	ext{th}}$ roots of unity?

Steve · Answer

z^4+z^2+1 = 0 (z^2-z+1)(z^2+z+1) = 0 one root is z = (1+√3 i)/2 = cis(pi/3) so, z^3 = -1 That means z^6 = 1

What is the smallest positive integer $n$ such that all the roots of $z^4 + z^2 + 1 = 0$ are $n^{\text{th}}$ roots of unity?