I'm working on a description of each of the subfields of a cyclotomic extension generated by a $p^r$-th root of unity, where $p$ is an odd prime number and $r\geq 1$ is a natural number for . My intention is to find a primitive element of each of the subfields of the extension.
By now, I have already studied the case in which $r=1$ by using Gaussian Periods, arriving to the conclusion that
$\textbf{Theorem:}$ Let $p$ be an odd prime number and $\zeta$ a $p$-th root of unity in $\mathbb{C}$. Let $K\subseteq\mathbb{Q}(\zeta)$ be a subfield and let $n:=[K:\mathbb{Q}]$ be the degree of the extension over $\mathbb{Q}$. Then, for any given $i\in\{0,\dots,n-1\}$, $K=\mathbb{Q}(\eta_i)$, where $\eta_i=\eta_i(n,\zeta,g)$ is the $i$-th $n$-th Gaussian period of $\zeta$ in relation to $g$, a generator of the multiplicative group $(\mathbb{Z}/p\mathbb{Z})^{\ast}$.
Also, I have shown a more general version of this theorem, for $r>1$, arriving to the conclusion that
$\textbf{Theorem:}$ Let $p$ be an odd prime number, $r\geq 1$ a natural number and $\zeta$ a $p^r$-th root of unity in $\mathbb{C}$. Let $K\subseteq\mathbb{Q}(\zeta)$ be a subfield such that the degree of the extension over $\mathbb{Q}$ is $[K:\mathbb{Q}]=p^{r-1}n'$ where $n'$ is a natural number such that $gcd(n',p)=1$. Then, for any given $i\in\{0,\dots,n-1\}$, $K=\mathbb{Q}(\eta_i)$, where $\eta_i=\eta_i(n,\zeta,g)$ is the $i$-th $n$-th Gaussian period of $\zeta$ in relation to $g$, a generator of the multiplicative group $(\mathbb{Z}/p\mathbb{Z})^{\ast}$.
The reason why I consider that the degree is $[K:\mathbb{Q}]=p^{r-1}n'$ is that there's a point in my proof where I have to use the fact that $p^{r-1}(p-1)/n\leq p-1< p$, thus the exponents $k$ of $\zeta$ satisfy $1\leq k<p$ and, therefore, the $k$'s are invertible.
My question is, if $[K:\mathbb{Q}]=p^{t}n'$, what happens when $t\neq r-1$?