For a smooth projective curve $C$ and $f,g \in k(C)^*$, we have $\text{div}(f)=\text{div}(g)$ iff $f=ag$ for some nonzero constant $a$. Is this still true for higher dimension smooth projective variety $X$ ? I believe this boils to : for all nonconstant $f \in k(X)$, $f$ must vanish somewhere on $X$? This sounds intuitive and trivial, but I can't find it in any reference. Any suggestion? Thank you.
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