The only two rational values for cosine and their connection to the Kummer Rings

Solution 1:

In a number field $K$, the norm of an element $N_{K/\mathbb{Q}}(a) = N(a)$ can be given various equivalent definitions, one of which is that it is the determinant of the linear map $x \mapsto ax$ acting on $K$ regarded as a vector space over $\mathbb{Q}$. If $\sigma_i : K \to \mathbb{C}$ denote the complex embeddings of $K$, then we also have

$$N(a) = \prod_i \sigma_i(a).$$

The norm is always rational.

If $K$ has degree $n$, then it has $n$ complex embeddings (for example by the primitive element theorem); in particular, fixing a basis of $K$ and expressing $a$ in it, the norm is a homogeneous polynomial of degree $n$. It is a quadratic form if and only if $n = 2$.

Now the degree of $\mathbb{Q}(\zeta_m)$ is equal to $\varphi(m)$, which is equal to $2$ if and only if $m = 3, 4, 6$. In other words, these are the only cyclotomic fields which give quadratic extensions. This is related to the crystallographic restriction theorem.

Yes, you can use cyclotomic fields to prove that $\cos \frac{2\pi}{m}$ is rational only when $m = 1, 2, 3, 4, 6$. Once you know that $\mathbb{Q}(\zeta_m)$ has degree $\varphi(m)$ (but this is not trivial), you can show that that $\mathbb{Q}(\zeta_m + \zeta_m^{-1})$ is a subfield of index $2$, hence is $\mathbb{Q}$ if and only if $\varphi(m) \le 2$.