For which values of $n$, the real part of the $n$-th root of unity is a quadratic irrational?

Question

For which values of $n$, the real part of the $n$-th root of unity is a quadratic irrational? That is, when is it a root of a quadratic polynomial with integer coefficients?

I believe that the answer is $n=3, 5, 6, 8, 12$, but I might be wrong. Any simple proofs?

Thank you very much.

Answer 1

Using Chebyshev polynomials, it is easy to check that $$\cos\frac{2\pi}{n}$$ is an algebraic number of $\mathbb{Q}$ with degree $\frac{\varphi(n)}{2}$, and the listed natural numbers are the only ones for which $\varphi(n)\in\{2,4\}$.