Complex Numbers: Solving Equations with z and zeta?

[lektor]

Hi, I have no clue how to approach this question, was in my last years final exams.

(z^2 + 1)^4 = 1

Find all solutions, where z is a complex number.

Tips please?

 

[arildno]

First off, write $1=e^{ik2\pi}$
where k is an integer.
Now, take the fourth root of the equation:
$$z^{2}+1=e^{i\pi\frac{k}{2}}$$
For how many choices of k will the right-hand side represent DISTINCT complex numbers?

 

[quasar987]

Supposing you have some experience solving equations of the type $z^n=a$ where a is complex...

Set $\zeta = z^2+1$. Then find all 4 solutions of $\zeta^4 = 1$. Then go back to $\zeta -1 = z^2$ and for all four $\zeta$ found, find the 2 solutions of z associated, for a total of 8.