# Irreducible polynomial over Dedekind domain remains irreducible in field of fractions

Let $\mathcal{O}$ be a Dedekind domain, $K$ its field of fractions. Suppose $f\in \mathcal{O}[X]$ is irreducible. Is it irreducible in $K[X]$?

The motivation for my question is that this is true for UFD’s, so it is natural to ask if it is still valid over an arbitrary Dedekind domain.

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This has been answered pretty exhaustively by Pete Clark on MO. The upshot is that as soon as the class number is not 1, the result does not hold.