If A $\subseteq B$ are domains and $B$ is integral over $A$, then is $Frac(B)$ algebraic over $Frac(A)$? Ia it because of the following:
Any element of $Frac(B)$ is of the form $b/t$ with $b,t \in B, t\neq 0$, and if $b, t$ are algebraic over $Frac(A)$ (which they are), then so is $b/t$ because $Frac(A) \subseteq Frac(A)(b,t)$ is an algebraic extension (and $b/t$ is in there).