Irreducible polynomial over Dedekind domain remains irreducible in field of fractions

Question

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.

score 2 · Accepted Answer · edited Apr 13 '17 at 12:58

2

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.

edited Apr 13 '17 at 12:58

Community

answered Nov 16 '12 at 13:06

Alex B.

1 Answers1