Let's consider $p(x)=x^p-1$, assuming $p$ as a prime number. Its degree over $\mathbb{Q}$ is $p-1$ and its Galois Group is $\mathcal{C}_{p-1}$, so there is a unique subextension of degree 2. Let's call $L$ this subextension. In which cases $\sqrt{p} \in L$?
For $p=5$ and $p=7$ it happens.