Quadratic Gauss Expression for primitive $7$th roots of unity

Question

For primitive $7$th root of unity $\omega$, calculate $|1+2\omega + 2\omega^2 + 2\omega^4|$.

Answer 1

Given that $w$ is a primitive 7th root of unity, we have $\overline w=w^6, \overline {w^2}=w^5, $ $\overline {w^4}=w^3$, $w^7=1$,

and $1+w+w^2+w^3+w^4+w^5+w^6=\dfrac{w^7-1}{w-1}=0$.

Therefore $|1+2w+2w^2+2w^4|^2=(1+2w+2w^2+2w^4)\overline{(1+2w+2w^2+2w^4)}$

$=(1+2w+2w^2+2w^4)(1+2w^6+2w^5+2w^3)=13+6w+6w^2+6w^3+6w^4+6w^5+6w^6$

$=7+6(1+w+w^2+w^3+w^4+w^5+w^6)=7.$