TITLE: Example of GCD in UFD that can't be expressed as linear combination
QUESTION [9 upvotes]: I proved that any two elements in PID have GCD and it can be expressed as linear combination of those two elements.
I know that even in case of UFD GCD exists but it may not be expressed as linear combination. Can anyone give me an example for which GCD is not expressed as linear combination?

REPLY [3 votes]: In $ℚ[X,Y]$ we have $\gcd (X,Y) = 1$, but $1 ≠ fX + gY$ for any $f, g ∈ ℚ[X,Y]$ and both of these assertions can be seen by looking at degrees in $X$ and $Y$.