Say, $C$ is a proper, normal curve over $k$. Why is $\mathrm{trdeg}_k(K(C))=1$ ?
Here, (curve) $\dim(C)=1$ means the topological dimension is $1$ and I know that it is equivalent to the fact that supremum of Krull dimensions of open affines in $C$ is $1$. Also I know that $K(C)=O_{C,\eta}$ where $\eta \in C$ is a generic point. But how do I proceed?