Find $c(x,y), d(x,y) \in F[x,y]$ such that $a(x,y)c(x,y) + b(x,y)d(x,y)$ is a non-zero polynomial in $F[X]$

Question

Let $F$ be a field. Let $a(x,y), b(x,y)$ be co-prime in $F[x,y]$.Prove that there exist $c(x,y), d(x,y) \in F[x,y]$ such that $a(x,y)c(x,y) + b(x,y)d(x,y)$ is a non-zero polynomial in $F[x]$.

Here’s my attempt: (For brevity I drop the arguments of the polynomials)

Let $R= F[x]$ and let $S$ be the field of fractions of $R$. Since $S$ is a field, $S[y]$ is a PID. Moreover $a, b$ remain co-prime in $S[y]$. Hence Bezout’s Lemma applies and we have $p,q \in S[y]$ such that $ap +bq=1$. Clearing denominators of $p,q$ gives the desired result!

Is this correct? Many thanks!

Answer 1

It looks good to me. Every step is clear.