How to calculate $z^4 + \frac1{z^4}$ if $z^2 + z + 1 = 0$?

Given that $z^2 + z + 1 = 0$ where $z$ is a complex number, how do I proceed in calculating $z^4 + \dfrac1{z^4}$?

Calculating the complex roots and then the result could be an answer I suppose, but it’s not quite elegant. What alternatives are there?

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From $z^2 + z + 1 = 0$, we have
$$z +\frac{1}{z}=-1.$$
Taking square, we get
$$z^2 +\frac{1}{z^2}+2=1,$$
which implies that
$$z^2 +\frac{1}{z^2}=-1.$$
I think you will know what to do next.

Essentially the same calculation also follows from the observation that $x^2+x+1=\phi_3(x)$ is the third cyclotomic polynomial. So $z^3-1=(z-1)(z^2+z+1)=0$, and hence $z^3=1$ for any solution $z$. Therefore
$$
z^4+\frac{1}{z^4}=z\cdot z^3+\frac{(z^3)^2}{z^4}=z+z^2=-1.
$$

$$\begin{align*}
z^4+\frac1{z^4}&=(-z-1)^2+\frac1{(-z-1)^2}\\
&=z^2+2z+1+\frac1{z^2+2z+1}\\
&=(-z-1)+2z+1+\frac1{(-z-1)+2z+1}\\
&=z+\frac1{z}=\frac{z^2+1}{z}=\frac{-z-1+1}{z}=-1
\end{align*}$$

Here is an alternative approach: let’s consider $z^{8}+1$ , and then divide by $z^{4}$.

By using geometric series, notice that $$z^{8}+z^{7}+z^{6}+z^{5}+z^{4}+z^{3}+z^{2}+z+1=\left(z^{2}+z+1\right)\left(z^{6}+z^{3}+1\right)=0.$$ Now, as $z^{2}+z+1=0$, we know that both $z^{7}+z^{6}+z^{5}=0$ and $z^{3}+z^{2}+z=0$, and hence $$z^{8}+z^{4}+1=0$$ so that $z^{8}+1=-z^{4}$. Thus, we conclude that $$z^{4}+\frac{1}{z^{4}}=-1.$$

Different people see different things in the relation $z^2+z+1=0$. Just as Jyrki did, I see the third cyclotomic polynomial. Its roots are $-1/2 \pm \sqrt{-3}/2$, the two primitive cube roots of unity. Call one of these $\omega$ and see that $\omega^3=1$, so that $\omega^4=\omega$ and $\omega^{-4}=\omega^2$. Their sum is $-1$, from the defining relation.

Hint $\ $ Exploit innate symmetry: for $\rm\ y = z^{-1} $ you know $\rm\ yz\ (=\: 1)\:$ and $\rm\ y+z\ (=\: z^{-1}\!+z\: =\: -1)\ $

Thus you know $\rm\ \ \ y^2 + z^2\ =\ (y\ +\ z)^2-\ 2\:(y\:z)$

hence you know $\rm\ y^4 + z^4\ =\ (y^2\! + z^2)^2 – 2\:(y\:z)^2$

For more on symmetric polynomials see the Wikipedia article on Newton’s identities.

$$z^2+z+1=0$$ Clearly $z \not= 0$ and therefore $z+\dfrac{1}{z}=-1.$
Note that $$z^{n+1}+\frac{1}{z^{n+1}}=\left(z+\frac{1}{z}\right)\left(z^n+\frac{1}{z^n}\right)-\left(z^{n-1}+\frac{1}{z^{n-1}}\right).$$
Therefore if we define the function $U:\Bbb{N}\cup\{0\} \to \Bbb{C}$ as $$U_n=z^n+\frac{1}{z^n}$$ then we have $U(0)=2,$ $U(1)=-1$ and $$U(n+1)+U(n)+U(n-1)=0\,\,\,\,\,\,\,\,\,\forall n \in \Bbb{N}.$$
Using this recurrence you can calculate $z^n+\dfrac{1}{z^n}$ for any $n \in \Bbb{N}.$

Alos you can follow this approach.