I have the following question:

Let $A$ be an integrally closed Noetherian domain, $K$ its field of fractions. let $L$ be a finite extension of $K$, and $B$ the integral closure of $A$ inside $L$. Is it true then that $B$ is finite over $A$?

If $L/K$ is separable, it is true (there is a usual proof with considering the non-degenerate bilinear form $tr(xy)$). What about a non-separable extension?

Thanks, Sasha