Let $p$ be an odd prime. Let $F$ be the splitting field of $x^p-1$ over $\mathbb{Q}$. Prove that there is a unique field $K$ bewtween $\mathbb{Q}$ and $F$ which is of degree 2 over $\mathbb{Q}$.
I know that $F = \mathbb{Q}(\xi_p)$ and $[F : \mathbb{Q}] = p-1$, which is divisible by 2. My intuition tells me that this field $K$ corresponds to the subgroup of order 2 of $D_n$, which would be generated by complex conjugation but I don't know how to go abour showing it.
I appreciate any help.
As you noted, the splitting field is $\mathbb{Q}(\zeta_p)$. The Galois group of $\mathbb{Q}(\zeta_p)/\mathbb{Q}$ is $(\mathbb{Z}/p\mathbb{Z})^{\times}$, which is cyclic. Therefore it has a unique subgroup of index $2$, and thus by Galois correspondence, there is a unique sub-extension of degree $2$.