A first part of a question on a practice exam asks whether there exist $a,b \in \mathbb{Z}$ such that $ap + bn = 1$, where $p$ and $n$ are integers such that $p$ does not divide $n$, and $p$ is prime.
So $gcd(p,n) = 1$, and so we can write $p = nq_1 + r_1$ where $r \neq 0$, then $n = q_2r_1 + r_2$, and carrying out the euclidean algorithm the procecess terminates at remainder $1$ and we can write it out backward.
I am then asked whether there are solutions $h,k \in \mathbb{Q}[x]$ such that $hf + kg = 1$ where $f$ is irreducible and does not divide g. I think that the division algorithm for polynomials would extend a similar argument here.
I am then asked the same question but considering $\mathbb{Q}[x,y]$ instead of $\mathbb{Q}[x]$, and I'm not sure where to proceed. I only understand that the ideals of this are not principal, but I'm not sure how to say that the division algorithm wouldn't apply here.