$\mathbb{Q}(\zeta_{15})$ is a field extension of $\mathbb{Q}$, where $\zeta_{15}$.I am trying to find the number of $i$ such that:
$\mathbb{Q} \subset L_i \subset\mathbb{Q}(\zeta_{15})$
Is there a way of using Galois theory to find this?
[Note: Sorry I have not provided much of my own working; I have not managed to find any solutions.]
You really have to look closely at the Galois group, $C_4\oplus C_2$, where I’m using $C_m$ for cyclic group of order $m$. It has three subgroups of order $2$ and three of order $4$. It would steal your fun for me to tell you what they all are.