I am trying to prove that $\Phi_m(x)$ is irreducible over $\Bbb Q(\zeta_n)$ if and only if $(m,n)\leq2$.
The left implication turns out to be somewhat easy since without loss of generality, $2\mid m$ and no higher power of $2$ divides $m$ and using the fact that $\Bbb Q(\zeta_{m/2})=\Bbb Q(\zeta_m)$ (this assumes the degrees aren't relatively prime to begin with, which I think is even easier).
My problem comes in doing the forward implication. I tried and tried without success, so I started trying to use the contrapositive, that is, if $(m,n)\geq3$, then $\Phi_m(x)$ is reducible over $\Bbb Q(\zeta_n)$. This hasn't proven fruitful yet, so if possible, will someone point me in the right direction?
First show that $\Phi_m(x)$ is irreducible over $\mathbb{Q}(\zeta_n)$ if and only if $[\mathbb{Q}(\zeta_m,\zeta_n):\mathbb{Q}(\zeta_n)]=\varphi(m)$. Then compute $[\mathbb{Q}(\zeta_m,\zeta_n):\mathbb{Q}(\zeta_n)]$ and the result will fall out.