# On finiteness of the integral closure of a subalgebra of an algebraic function field of one variable

This question is motivated by this.

Let $k$ be a field. Let $K$ be a finitely generated extension field of $k$ of transcendence degree one. Let $A$ be a subring of $K$ containing $k$. Suppose that $A$ contains a transcendental element over $k$. Let $B$ be the integral closure of $A$ in $K$. Is $B$ finitely generated as an $A$-module?

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