Possible solutions for complex equations I have trouble solving this complex equation:
If $$z^4 +z^3 + z^2 + z + 1 = 0 $$ and $$u = z + z^{-1}$$ find all the possible values for u . I have tried substituting u into the equation in two different ways, finding that $$ u = \frac{-1}{z^3 + z^2}$$ and $$ u = \frac{z^4 -1}{z^3 - z} $$
but I'm not sure where to go from here. How many possible values should I expect to find for u? Is it possible to find the values of z with the information I have?

We have 
\begin{eqnarray*}
u &=& z+\frac{1}{z} \\
u^2 &=& z^2+\frac{1}{z^2}+2. \\
\end{eqnarray*}
So 
\begin{eqnarray*}
u^2+u-1=0   \\
u = \frac{-1 \pm \sqrt{5}}{2}. \\
\end{eqnarray*}
Now you just need to solve the quadratic 
\begin{eqnarray*}
z+\frac{1}{z}= \frac{-1 \pm \sqrt{5}}{2}. \\
\end{eqnarray*}
