complex numbers equation deg 4

Question

Find the sum of squares of elements of set $A$ if: $$A=\Big{\{}\big|z^n+\frac{1}{z^n}\big|;\:n\in\mathbb{N},\:z\in\mathbb{C},\: z^4+z^3+z^2+z+1=0\Big{\}}.$$

Answer 1

Since $z^5=1$ (so $|z|=1$) it is not difficult to find $$A=\{ 2,|z+1|,|z^2+1|\}$$ If we devide $z^4+z^3+z^2+z+1=0$ with $z^2$ and put $t=z+{1\over z}$( so $t$ is a solution of $t^2+t-1=0$), then $$|z+1|^2 = |z^2+2z+1| = |z||z+2+{1\over z}| = |t+2|$$ and $$|z^2+1|^2 = |t|^2$$ so we have $$S= 4+|z+1|^2+|z^2+1|^2= 4+|t+2|+|t|^2 $$

1. case If $t={\sqrt{5}-1\over 2}$ we get $S= 7$

2. case If $t={-\sqrt{5}-1\over 2}$ we also get $S= 7$

So $$\boxed{S=7}$$