I don't know how to prove the following proposition.
If two varieties $X$ and $Y$ are irreducible, a morphism $\phi: X \rightarrow Y$ is dominant and finite, then $K(X)$ is a finite algebraic extension of $\phi^*K(Y)$.
Here, $\phi^*: K[Y] \rightarrow K[X]$ sends a function $f \in K[Y]$ to $f \circ \phi \in K[X]$. A morphism $\phi$ of irreducible varieties is called dominant if $\phi(X)$ is dense in $Y$. It is called finite if $K[X]$ is an integral over $\phi^*K[Y]$.
Is the following more general statement true?
$R_1$ is a domain, which is integral over its subdomain $R_2$. $F_1$ and $F_2$ are respective fields of fractions. Then $F_1/F_2$ is a finite algebraic extension.
This extension must be algebraic. But I think the finiteness is not so obvious.