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

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?

• Your special case $p=3$ though correct, it is over simplified. "A group order 2 has a subgroup of index 2, namely the trivial subgroup" is not a hard statement. Try $p=5$ at least. – P Vanchinathan Jul 3 at 0:43
• It isn't hard to compute the Galois group very explicitly in this case. Hint: a permutation is determined by where you send your $a$, and you can send it to any root. What does this imply about the Galois group? – RghtHndSd Jul 3 at 2:02