Roots of unity over $\mathbb{Q}$

Question

I want to show the following proposition from Algebra, Hungerford V.8.9.

If $n > 2$ and $\xi$ is a primitive $n$th root of unity over $\mathbb{Q}$, then $[\mathbb{Q}(\xi + \xi^{-1}) : \mathbb{Q}]= \frac{\varphi(n)}{2}$, where $\varphi(n)$ denotes the Euler's totient or phi function.

Thanks in advance.

Answer 1

Hint: Under the Galois correspondence prove that $\mathbb{Q}(\zeta+\zeta^{-1})$ corresponds to the subgroup $\langle -1 \rangle$ of order $2$ of $\mathrm{Gal}(\mathbb{Q}(\zeta)/\mathbb{Q}) \cong (\mathbb{Z}/n)^*$. Since $-1$ corresponds to the complex conjugation, you have to prove $\mathbb{Q}(\zeta+\zeta^{-1}) = \mathbb{Q}(\zeta) \cap \mathbb{R}$.