Let $L/K$ be an algebraic extension, and let $a,b \in L$. Suppose that $f(X,Y) \in K[X,Y]$ is a polynomial such that $f(a,b)=0$.
Is it true that $$f(x,y)= P_{\text{min},a,K}(x) P(x,y) + P_{\text{min},b,K}(y) Q(x,y) $$ for some polynomials $P,Q \in K[X,Y]$ ?
I know that this is true if we consider instead $f(X) \in K[X]$, because $K[X]$ is an euclidean ring. My idea was to consider $K(X)[Y]$ which is euclidean, and then to come back to $K[X,Y]$, but this is probably not the best idea.
Thank you for your help!