When $(\mathbb{Q}(\zeta_n) : \mathbb{Q}) = 2$?

Question

maybe this is a stupid question . Anyway, when $(\mathbb{Q}(\zeta_n) : \mathbb{Q}) = 2$, i.e., $\phi(n) = 2$ (where $\phi$ is the Euler's totient function)?

Thanks in advance.

Answer 1

Hint: if $n=p_1^{a_1}p_2^{a_2}\cdots p_k^{a_k}$, then $$\varphi(n)=(p_1-1)p_1^{a_1-1}\cdots(p_k-1)p_k^{a_k-1}$$ (this formula is an easy restatement of the one listed on Wikipedia). Thus, if $\varphi(n)=2$, could $n$ be divisible by $5$, for instance? What can you deduce about the possible powers of $2$ or $3$ going into $n$?