In a PID, we know that if we know that the coefficients of an irreducible polynomial don't have any common factor. Is the same true in a Dedekind domain ?
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or Are there irreducible polynomials of the form $a_n x^n + a_{n-1} x^{n-1} + \ldots + a_0$ over a Dedekind domain $R$ such that $a_i \in \mathcal{P}$ for all $i$?
Note: There are such linear polynomials. See comments.