Does a rational function $\phi$ on a smooth projective algebraic curve $F$ over a algebraically closed field $K$ always have a representative $\frac{f}{g}$, where $f$ and $g$ are polynomials without common zero on $F$ ?
I think that it is a consequence of Max Noether's fundamental theorem ($AF + BG$ theorem), which would in this sense be the analogon of the fundamental theorem of algebra (simplification of rational functions over $\mathbb{C}$).