Let $n>2$ prove that $[\mathbb Q(z+\frac{1}{z}):\mathbb{Q}]=\frac{1}{2}\varphi(n)$ for every primitive $n$-th root of unity $z$

Question

Let $n>2$ prove that $\text{deg}(m_{z+1/z}):=[[\mathbb Q(z+\frac{1}{z}):\mathbb{Q}]=\frac{1}{2}\varphi(n)$ for every primitive $n$-th root of unity $z$

So, since I know that with $z$ being a primitive $n$-th root of unity $1/z$ is one aswell. I suppose that for a correct proof I would have to look at the factors in

$\varphi(n)=\prod\limits_{i=1}^{r}(p_{i}-1)p_{i}^{k_{i}-1}$ with $n=p_{1}^{k_{1}},...,p_{r}^{k_{r}}$

However I'm really stuck with this problem.