I've been working on some problems regarding polynomials, and I ended up with the following question:
Suppose that $\zeta_{p^n}$ is a root of unity of order $p^n$, and consider the cyclotomic extension $\mathbb{Q}[\zeta_{p^n}]|\mathbb{Q}$. The corresponding Galois group has order $\phi(p^n) = (p-1)p^{n-1}$. Therefore, there must be a subgroup of order $p-1$, that should correspond, by the correspondence given in the Fundamental Theorem of Galois Theory, to some subfield of $\mathbb{Q}[\zeta_{p^n}]$.
I suspect that this subfield is exactly $\mathbb{Q}[\zeta_{p^n}^p] \cong \mathbb{Q}[\zeta_{p^{n-1}}]$, but I don't know any reference of this fact. Is there a reference for this? Or if I am wrong, wich subfield should the group correspond to?
Any help would be appreciated.
If $p\ne2$, the Galois group $G$ of the cyclotomic extension is cyclic. It's $\Bbb Z_{p^n}^×$. There's one cyclic subgroup of order $p-1$, since $(p-1)\mid\varphi(p^n)$. Call it $N\triangleleft G$.
$N$ corresponds to an extension $K$ of $\Bbb Q$ of degree $[G:N]=p^{n-1}$.
But $[\Bbb Q(\zeta_{p^{n-1}}):\Bbb Q]=\varphi (p^{n-1})$, since it's a cyclotomic extension.
Since, $\varphi (p^{n-1})\ne p^{n-1}$, this is a different extension.
The extension you want is $K=\Bbb Q^N$, the fixed field under the action of $N$.
$K$ is not a cyclotomic extension in this case.