Greetings Mathematics Community.
I am having much difficulty in solving the following problem:
If $m\equiv 2$ (mod 4), show that $\mathbb{Q(\zeta_m)}=\mathbb{Q(\zeta_{\frac{m}{2}})}$ where $\zeta$ is a primitive root of unity. I know that $\zeta_m=e^{\frac{2\pi i}{m}}$ and by Euler's equation, we have $\cos(\frac{2 \pi }{m})+i\sin(\frac{2\pi }{m})$.
I have tried to compare $[\mathbb{Q(\zeta_{\frac{m}{2}})}:\mathbb{Q}]$ with $[\mathbb{Q(\zeta_m)}:\mathbb{Q}]$ by substituting $\frac{m}{2}$ in Euler's equation to get $\cos(\frac{4 \pi }{m})+i\sin(\frac{4\pi }{m})$. My intuition is telling me that I should somehow use the given fact that $m\equiv 2$ (mod 4), but I am not sure how.
Furthermore, I do know that this extension degree can be found using Euler's $\phi-$function on $m$. But I am unsure about how to apply it in my situation.
As always, any help is greatly appreciated. Thanks in advance.