Let $f:X\rightarrow Y$ be a dominant morphism between irreducible varieties over an algebraically closed field $k$.
When is $k(X)$ algebraic over $k(Y)$? Is there an if and only if criterion? What if $f$ is even bijective? From what I've found it should be enough that $f$ is finite. Is there a weaker criterion?
$f$ being dominant implies that the corresponding ring homomorphism is injective. Thus we have an inclusion of integral domains over $k$, $A\hookrightarrow B$ and the function fields in question are the fraction fields of these domains. Obviously, we have $frac(A)\hookrightarrow frac(B)$.
Assuming that $A,B$ are finitely generated as algebras over $k$, we have a theorem which says that the dimension of A is equal to the transcendence degree of frac(A) over k. The same goes for $B$, of course. Clearly, the bigger field is algebraic over the smaller one if and only if both have the same transcendence degree over $k$, meaning $\dim A=\dim B$, or equivalently, $\dim X=\dim Y$.