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Suppose $R$ is an integral domain with quotient field $K$ and $f \in R[x]$ is an irreducible polynomial.
Under what conditions on $R$ and $f$ can we conclude that $f$ is irreducible in $K[x]$?

It is well-known that if $R$ is factorial, then this holds for any polynomial. It is perhaps a bit less well-known that factorial can be weakened to GCD domain.
Obviously, if $R$ is integrally closed, then this will hold for any monic polynomial.

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    $\begingroup$ I think you may be confused. Let $c$ be the gcd of the coefficients of a nonzero polynomial $f \in R[X]$, where $R$ is a UFD. Then $f=cg$ where $g \in R[X]$ is primitive, and $fK[X] = gK[X]$. So $K[X]$ really doesn't care whether a polynomial is primitive over $R$; it just factors out the content. It's always true that if $f$ is a polynomial over an integrally closed domain, and if $f$ is irreducible over $R$, then it is irreducible over $K$. I think so anyway. The "converse" that holds over a UFD is that if $f$ is primitive over $R$ and irreducible over $K$, then it is irreducible over $R$. $\endgroup$
    – Neil Epstein
    Commented Feb 11, 2018 at 22:19
  • $\begingroup$ @NeilEpstein Dear Neil, thank you. You're right, I was confused, sorry. I have edited the question accordingly, I hope that it makes sense now. $\endgroup$
    – Lukas Heger
    Commented Feb 11, 2018 at 22:35
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    $\begingroup$ @NeilEpstein You must rule out the case where $f$ is an irreducible constant. $\endgroup$
    – Laurent Moret-Bailly
    Commented Feb 12, 2018 at 11:45

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