Map from algebraic curve to $\mathbb{P}^1$ an isomorphism?

Question

Let $C$ be a connected algebraic curve proper over a field $k$. Let $P$ and $Q$ be distinct closed points of $C$. Let $ f$ be a meromorphic function on $C$ such that $div(f)=P-Q $. By valuative criteria for properness $ f$ induce a morphism of schemes $f:C \to \mathbb{P}^1_k$. Is this morphism an isomorphism? Do I need to assume $k$ is algebraically closed?