Working on the following practice problem in studying for qualifying exams:
Let $\mathbb{C}(T)$ be the field of rational functions in $T$ over the complex numbers. Let $n$ be a positive integer and let $\zeta$ be a primitive $n$-th root of unity.
(a) Let $\sigma$ be the automorphism of $\mathbb{C}(T)$ that maps each rational function $f(T)$ to $f(\zeta T)$. Show that the fixed field of $\sigma$ is exactly $\mathbb{C}(T^n)$.
$\rightarrow$ This part is pretty straightforward, but essentially the argument shows that $\sigma$ fixes $x^k$ if and only if $n | k$. The portion I am struggling with is the following extension
(b) Find all intermediate fields $K$ between $\mathbb{C}(T^n)$ and $\mathbb{C}(T)$.
$\rightarrow$ My intuition for this is that the intermediate fields will be in correspondence with $\mathbb{C}(T^k)$ where $k | n$, but I am struggling to show the proper Galois group and its subgroups.