I'm confused.
Let $X$ be a curve over a field $k$. Let $K= K(X)$ be its function field. Then, $K(X)$ is a field. Each non-constant morphism $f:X\to \mathbf{P}^1_k$ gives a field extension $K(\mathbf{P}^1_k) = k(t) \subset K(X)$ of degree $\deg f$.
How is it possible that $\deg f$ can take infinitely many values? Aren't both fields fixed?
Maybe I'm not understanding the situation fully. A rational function $f$ on $X$ gives a field extension $k(t) \subset k(t,f)$ of degree $\deg f$. Maybe $k(t,f) \not= K(X)$? No...
Can somebody explain?