TITLE: Is Gauss's lemma valid for polynomials with coefficients in a GCD domain?
QUESTION [9 upvotes]: Wikipedia's proof of Gauss's lemma requires this theorem:

If $(C \mid S\cdot T) \land \lnot \operatorname{invertible}(C)$, $C$ has a non-invertible divisor in common with at least one of $S$ and $T$.

I can prove it for a Bézout domain, but not a GCD domain. Can anybody help me?
Original text:

If the contents $c = c(ST)$ is not invertible, it has a non-trivial divisor in common with the leading coefficient of at least one of $S$ and $T$ (since it divides their product, which is the leading coefficient of $ST$).

REPLY [7 votes]: FYI: a proof of Gauss's Lemma for GCD domains is given in Section 15.5 of these notes.  (I don't claim any superiority to any other proofs you may have seen or discovered...)