Algebraic element becoming transcendental over residue field?

Question

Let $A\subseteq B$ be an algebraic extension of integral domains, i.e., every element of $B$ satisfies a nonzero polynomial over $A$ (not necessarily monic). Let $m,n$ be maximal ideals of $A,B$ respectively, such that $m=n\cap A$. Can it happen that the induced natural inclusion of fields $A/m\hookrightarrow B/n$ is not algebraic? By Hilbert's Nullstellensatz, it is not possible if $B$ is a finitely generated $A$-algebra. But I am interested in valuation rings. Is it possible that, for some element of $B$, all relations `vanish' when we go mod $m$?

Answer 1

Let $k$ be a field, $A:=(k[Y,Y/X])_{(Y,Y/X)}$ and $B:=k[X,Y]_{(Y)}$. Then $A\subseteq B$ is a local inclusion, which is also an algebraic extension as $Q(A)=Q(B)=k(X,Y)$. But the residue fields of $A,B$ are $k$ and $k(X)$ respectively.