Here is the problem at hand:
Let $A$ be a UFD which is a $\mathbb{Q}$-algebra. Let $K$ be the fraction field of $A$. Let $L$ be a quadratic extension field of $K$. Show that the integral closure of $A$ in $L$ is a finitely generated free $A$-module.
We can deduce that the characteristic of $K$ is 0, which implies that $L/K$ is separable. Since $L$ is a degree 2 extension, $L/K$ is normal, implying that $L/K$ is Galois. Furthermore, $A$ being a UFD implies that $A$ is integrally closed in $K$. With these two pieces of information, I feel like there are plenty of tools in Galois theory and commutative algebra at my disposal, but I'm not sure what to use. Any small hints are appreciated!