The splitting field of $x^p-1 \in \mathbb{Q}[x] $ where p is odd prime, contains a unique subfield of index 2

Question

I took particular case when p=3 then I find that $\mathbb{Q}$ is subfield of index 2 in this particular case.

I also find that splitting field of given polynomial is $\mathbb{Q(a) }$ where a is pth root of unity such that a is not 1.

How can I find unique subfield of index 2 in general?