I have a problem figuring out how exactly I find the cube roots of a cubic with complex numbers.
I need solve the cubic equation $z^3 − 3z − 1 = 0$.
I've come so far as to calculate the two complex roots of the associated quadratic but then I'm stuck. I've got the solutions here and my lecture notes, have a look at this:
What I don't understand is how you go from $e^{i\pi/3}$ to $e^{i\pi/9}$. Because that root, as I understand it, should be the two conjugates roots added together which I believe do not add up to $e^{i\pi/9}$. What's the step going on here?
Any help is much appreciated!