Suppose $B$ lies over $A$, $A,B$ integral domains. Does this tell us anything about which homomorphisms from $A$ to a field we can extend to $B$?
We say $B$ lies over $A$ if, for any prime ideal $\mathfrak a$ of $A$, there is a prime ideal $\mathfrak b$ of $B$ such that $A\cap \mathfrak b = \mathfrak a$.
We do have a theorem telling us that if $B$ is integral over $A$, and $\Omega$ is some algebraically closed field, we can extend a map $A \to \Omega$ to $B \to \Omega$. Are there other results telling us in which ways we can extend a map from $A$ to a field on the basis of knowing $B$ lies over $A$?