I have the 16-th primitive root of unity $\zeta$ and the extension $\mathbb Q(\zeta):\mathbb Q$, I want to find the $L$ subfields of this extension for which $[L:\mathbb Q]=2$.
Could anyone tell me what these subfields are and what their structure is? I've looked for examples online that manipulate the elements but I found them confusing, could I use the fact that I know $\zeta^2$ is a primitive 8-th root of unity and $\zeta^4$ is a primitive 4-th root of unity?
Help appreciated!