Let $X \to Y$ be a cyclic cover of algebraic varieties, with $Y$ smooth and $X$ normal, say over the complex numbers. Let $G$ denote the Galois group and let $\chi$ be a character. By Kummer theory (aka Hilbert 90), the extension $K(X) / K(Y)$ is generated by a function $f$ such that $f^\sigma / f=\chi(\sigma)$.

Is it true that $f$ does not vanish outside the ramification locus?

The picture I have in mind is $y^n=(x-a_1)\cdots (x-a_r)$ where this is the case, but maybe more complicated things can happen.