Find a positive integer $n$ such that there is a subfield of $\mathbb{Q}_n$ that is not a cyclotomic extension of $\mathbb{Q}$

Question

I'm in trouble with this simple exercise. If we denote $\mathbb{Q}_n:=\mathbb{Q}(\omega)$, where $\omega$ is a primitive $n$th of unity in $\mathbb{C}$, can we find a positive integer $n$ such that there is a subfield of $\mathbb{Q}_n$ that is not a cyclotomic extension of $\mathbb{Q}$?

I don't know if this is precisely simple, I've thought a lot and nothing comes to mind. Can you give me a hand please? I'm sure hints will suffice. Thanks.

Answer 1

Hint: We have $[\mathbb{Q}_n:\mathbb{Q}] = \varphi(n)$, which tends to be even.