<s>[INST] find all complex solutions of $z^{12}-z^6-2z^3+2=0$ How could I find all the complex solutions to: $z^{12}-z^6-2z^3+2=0$
I tried substituting $y=z^3$ to get a simpler equation and I managed to get $y-1=0$ or $y^3+y^2-2=0$, but finding all complex solutions of the latter equation was also complicated.
 [/INST] The equation $y^3+y^2-2$ has an obvious root: $y=1$, and it factorises as 
$y^3+y^2-2=(y-1)(y^2+2y+2)=(y-1)((y+1)^2+1)$, hence the initial equation is
$$y^4-y^2-2y+2=(y-1)^2((y+1)^2+1)=0$$
The solutions are 
$$y=1,\quad y=-1\pm\mathrm i.$$
There remains to compute the cubic roots of these numbers.
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