Conditions when $F_1=F_2$ given $\mathbb Q\subset F_1 , F_2\subset \mathbb Q(\zeta_n)$ and $[F_1: \mathbb Q]=[F_2:\mathbb Q]$ Suppose that $F_1, F_2$ are fields so that $\mathbb Q\subset F_1 , F_2\subset \mathbb Q(\zeta_n)$ and $a:=[F_1: \mathbb Q]=[F_2:\mathbb Q]:=b$.
If $n$ is prime, then I want to show that $F_1=F_2$.
I know that $[\mathbb Q(\zeta_n): \mathbb Q]=\phi(n)=n-1$ in this case and that $F_1, F_2$ correspond to cyclic subgroups of Gal$(\mathbb Q(\zeta_n)/\mathbb Q)$. Thus $a=b$ divides $n-1$. But how do I finish from here?
Also, I don't this will hold if $n$ is nonprime. But what is a counterexample?  
 A: Note that $G:= \text{Gal}(\mathbb{Q}(\zeta_n)/\mathbb{Q})$ is cyclic, so it has exactly one subgroup of any order dividing the order of $G$. Your hypothesis then implies that $\text{Gal}(\mathbb{Q}(\zeta_n)/F_1) = \text{Gal}(\mathbb{Q}(\zeta_n)/F_2)$, so $F_1 = F_2$ by Galois correspondence.
A: The Galois group is cyclic, so the result follows from the Galois
correspondence plus the fact that a cyclic group has at most one
subgroup of any given order.
In general, consider $n=p_1\cdots p_k$ where the $p_i$ are distinct
odd primes. The Galois group is a product of cyclic groups of orders
$p_i-1$ and so will have many order $2$ subgroups.
A: This is not the kind of answer you are seeking, but it is a bit long for comment.

In the final chapter of Disquisitiones Arithmeticae Gauss describes the cyclotomic field extension $\mathbb{Q} (\zeta_{n})\supseteq \mathbb{Q} $ and its subfields for the case when $n$ is prime. Using his theory of periods (sums of particular $n$'th roots of unity based on a primitive root of $n$) he showed that for each divisor $m$ of $n-1$ there is exactly one subfield $K $ with $\mathbb{Q} \subseteq K\subseteq \mathbb{Q}(\zeta_{n}) $ and $[K:\mathbb {Q}] =m$. It is interesting to see Gauss technique which uses primitive roots of $n$ (which are sort of used to find the generators of the cyclic group $\text{Gal} (\mathbb {Q} (\zeta_{n}) /\mathbb{Q}) $) without any knowledge of group theory. 
You may have a look at Gauss theory of periods in this post and the next which gives explicit construction of these subfields.
