I'm reading a book on algebraic curves, and at one point it says that if C is a smooth curve and f belongs to K(bar)(C)* for perfect field K, and if div(f)=0, then f has no poles. It's my understanding, that div(f) is the sum of order of f over various points P in C. So isn't it possible to have a function f with a pole and a zero of the same order at two different points P, and thus wouldn't the order still be 0?
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