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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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up vote 2 down vote accepted

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.

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