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. 



Kummer theory and ramified covers
The answer is *no* in general, as shown by the following example.


Let $\pi \colon X \to Y$ be an étale (i.e., unramified) double cover of projective curves over $\mathbb{C}$, with $g(X) \geq 3$ (for instance, we can take $g(X)=3$, $g(Y)=2$).


Then the ramification locus is empty. However, the generator $f$ of the $\mathbb Z / 2 $-extension $K(X)/K(Y)$ must necessarily vanish somewhere on $X$.


Indeed, since $X$ is compact, there exist no non-constant holomorphic functions on it, hence $f$ has at least one pole. But the Residue Theorem implies that the number of zeroes of $f$ equals the number of poles (if counted with multiplicities), so $f$ has at least one zero on $X$. 


