Possible solutions for complex equations

Question

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?

Answer 1

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*}

Answer 2

HINT

We have that for $z \neq 1$

$$z^4 +z^3 + z^2 + z + 1 = \frac{z^5-1}{z-1}$$

Answer 3

Notice that $z^4 + z^3 + z^2 + z + 1 = (z^5 - 1)/(z-1)$. So the solutions to the top equation are all the $5^{th}$ roots of unity except for $1$.