When solving Diophantine equations, often I pass to a number field $K$ and hope that the algebraic integers $O_K$ have unique factorization.
Suppose that $O_K$ is not a UFD. Is it possible that there exists a field extension $L \supset K$ such that $O_L$ has unique factorization? Does such an $L$ always exist? Is there a method to find such an $L$ when it exists? I have tried to construct examples, but the $L$'s always end up too large for me to compute the class number by hand.