Is the ideal $(2,3)$ principal in $\mathbb{Q}[x]$? I believe that I have shown that $(2,3)$ is non-principal in $\mathbb{Z}[x]$. My outline goes something like this: Assume that $(2,3) = (f(x))$ then $f(x)$ divides 2 and 3, that is 2 = $f(x)g(x)$ and 3 = $f(x)q(x)$, so the sum of the degrees of $f(x)$ and $g(x)$ is 0 and the same is true for $f(x)$ and $q(x)$. Let $f(x)=c$ where $c$ is a constant. This would then mean that $(2,3) = (\pm 1) = \mathbb{Z}[x]$, a contradiction. 
I was wondering whether this argument might work for the case of $\mathbb{Q}[x]$ but it seems that you could form any polynomial from $\mathbb{Q}[x]$ using the generator $(2,3)$. 
 A: $\mathbb{Q}[x]$ is a Euclidean Domain so all ideals $\mathbb{Q}[x]$ are principal.
A: Note  $\ $ When considering the persistence of coprimality (comaximality) it is essential to consider the persistence of $1$, i.e. that the (inclusion) homomorphism preserves $1$.$\ $ If  $\rm\:(r, s) = (1)\:$ in $\rm R$ then this persists in any superring $\rm S \supset R$ that has the same $1$, but it may fail if not, e.g. $\:3-2 = 1$ $\Rightarrow$ $(3,2) = (1)$ does not persist when embedded in $\mathbb Z^2$ via $\rm\:n\to (n,0)$ since in the superring it becomes $(3,0)-(2,0) = (1,0),$ but $(1,0)\ne (1,1) = 1_{\mathbb Z^2}.\:$ So comaximality does not persist in this case. Occasionally even experienced mathematicians make serious errors by overlooking this subtlety.
A: Note that $(2,3) = (1)$ in $\mathbb{Z}$, hence $(2,3)=(1)$ in any larger ring (that has the same unity) as well, in particular in $\mathbb{Z}[x]$ and in $\mathbb{Q}[x]$. There is no contradiction, since in fact $(2,3)$ does contain $1$: $(2,3)$ contains the difference of $3$ and $2$, after all.
On the other hand, $(2,x)$ is non-principal in $\mathbb{Z}[x]$.
More generally: if $D$ is a Unique Factorization Domain that is not a field, and $p$ is a prime of $D$, then $(p,x)$ is not principal in $D[x]$. 
On the other hand, if $D$ is a field, then every ideal of $D[x]$ is principal, since $D[x]$ is a Euclidean domain.
