The author proves that whenever $n$ is prime, the Galois group of $L:K$ is indeed abelian, yet it seems to me that the argument could be extended to all natural numbers $n$.
The author's proof:
The argument extended for all natural numbers $n$:
The zeroes of $t^n -1$ are the $n$th roots of unity, namely $e^{k2 \pi i/n}$ for $k = 0, 1, ..., n-1$. These roots are all different and form a group $G$ under multiplication where $e^{2 \pi i/n}=\epsilon$ generates $G$, thus $G$ is cyclic and abelian. Furthermore $L=K(\epsilon)$. The argument follows as before.
My concern is that the arguments are practically equal, so I don't understand why the author would only prove the theorem for the case where $n$ is prime, such idea makes me feel like I'm missing something.
Can the theorem actually be extended for all natural numbers or is my proof flawed?
I would really appreciate any help/thoughts.