<s>[INST] Find all five solutions of the equation $z^5+z^4+z^3+z^2+z+1 = 0$ $z^5+z^4+z^3+z^2+z+1 = 0$
I can't figure this out can someone offer any suggestions?
Factoring it into $(z+1)(z^4+z^2+1)$ didn't do anything but show -1 is one solution. 
I solved for all roots of $z^4 = -4$ but the structure for this example was more simple.
 [/INST] $$z^5+z^4+z^3+z^2+z+1 = 0 \Longleftrightarrow$$
$$(z+1)(z^2-z+1)(z^2+z+1) = 0 \Longleftrightarrow$$
$$z+1=0 \vee z^2-z+1=0 \vee z^2+z+1=0 \Longleftrightarrow$$
$$z=-1 \vee z=\frac{1\pm\sqrt{-3}}{2} \vee z=\frac{-1\pm\sqrt{-3}}{2} \Longleftrightarrow$$
$$z=-1  \vee z=\frac{1\pm i\sqrt{3}}{2}  \vee z=\frac{-1\pm i\sqrt{3}}{2} $$
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