Prove that $\mathbb{Q}(\zeta_{2^{n+2}} + (\zeta_{2^{n+2}})^{-1})$ is a cyclic field extension of $\mathbb{Q}$ of degree $2^n$.

Question

For a Galois theory course, I need to prove that $\mathbb{Q}(\zeta_{2^{n+2}} + (\zeta_{2^{n+2}})^{-1})$ is a cyclic field extension of $\mathbb{Q}$ of degree $2^n$. Constructing the minimal polynomial does not seem the way to go, so what would be the best approach?

Answer 1

You can do it with minimal polynomials: We have $x=\zeta_{2^{n+2}}+(\zeta_{2^{n+2}})^{-1}=2\cos(\pi/2^{n+1})$, so $T_{2^n}(x/2)=\cos(2^n\cdot\pi/2^{n+1})=0$, where $T_m$ is the Chebyshev polynomial of the first kind, etc.

However, it is easier if you just use the fact the cyclotomic extension $\mathbb{Q}(\zeta_{2^{n+2}})/\mathbb{Q}$ has degree $\phi(2^{n+2})=2^{n+1}$.

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Edit: The automorphisms are $\cos(\pi/2^{n+1})\mapsto\cos(k\pi/2^{n+1})$ where $k$ is odd. Recall $(\mathbb{Z}/2^{n+2}\mathbb{Z})^*=C_2\times C_{2^n}$, the $C_{2^n}$ being generated by $5$, so it suffices to show $5^{2^{n-1}}\not\equiv\pm 1\pmod{2^{n+2}}$, which you can find a proof here