LEC ALG3 12

gcd not defined in power series ring

If $R$ is a UFD, when is primitive $f(x)\in R[x]$ irreducible in $R[x]$?
Since $R$ is an ID, its ring of fractions $Q$ is a field, so irreducible elements are prime.
$f(x)$ is irreducible in $R[x]$ $⟺$ $f(x)$ is irreducible in $Q[x]$.

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[!Lemma]
Let $R$ be a UFD and $f(x)\in R[x]$ be a primitive polynomial. Then $f(x)$ is irreducible in $R[x]$ $⟺$ $f(x)$ is irreducible in $Q[x]$.

[!Proof]-
$(⟹)$ done

$(⟸)$ obvious.

[!Theorem]
If $R$ is a UFD, then $R[x]$ is a UFD.