I am a bit unsure as to how to find sub fields of field extensions of the rationals containing primitive roots. A specific question which I am stuck on is find all subfields $K$ of $\mathbb{Q}(\zeta_{16})$, $\zeta_{16}$ a root of unity of order sixteen, such that $[K:\Bbb{Q}]=2$.
I have attempted a similar question: find all subfields $K$ of $\Bbb{Q}(\zeta_{133})$ such that $[K:\Bbb{Q}]=3$. I was able to get a bit further with the Question by splitting up the galois group $Gal(\Bbb{Q}(\zeta_{133})/\Bbb{Q})$ into $Gal(\Bbb{Q}(\zeta_{7})/\Bbb{Q})\times Gal(\Bbb{Q}(\zeta_{19})/\Bbb{Q})$ and looking for subgroups of order 36 (=108/3) which I would then use to find the fields but I was wondering whether there was a more sensible way to do this? Possibly a way that would work also for the first question where you can't split into coprime factors?
Thanks for your help, Harriet