One of the exercises in Algebra Chapter 0 is to prove Fermat's Last Theorem holds in $\mathbb{C}[x]$. That is, if $f, g, h \in \mathbb{C}[x]$ are non-constant polynomials and $n \geq 3$, then $f^n + g^n \neq h^n$, or whatever equivalent formulation you prefer.
The exercise is well hinted and I think I can complete it alone, but there's one detail that is giving me problems. We have assumed that $f, g,$ and $h$ are all coprime, and have shown that
$$f^n = \prod_{i = 1}^n (h - \zeta^i g)$$
where $\zeta$ is an $n$th primitive root of unity. The hint now says "use that $\mathbb{C}[x]$ is a UFD to prove that $h - \zeta^ig$ is an $n$th power". That is, show that $h - \zeta^ig = a^n$ for some complex polynomial $a$. This is where I'm stuck.
I know I can break up $f$ into linear factors, $f_1, f_2, \dots, f_n$, so I assume that $h - \zeta^ig = f_r^n$ for some appropriate $f_r$. However, I can't see why $h - \zeta^ig$ might not be some other partition of the factors of $f^n$. I've tried looking up some other proofs of this fact, but unfortunately the sources I found simply say "because $\mathbb{C}[x]$ is a UFD, $h - \zeta^ig$ is an $n$th power", which is not very helpful.