I know that similar questions have been asked often. I have had a look at many of them, and I have found the following claim over polynomials in two variables in an answer here:
$$x-(by+c) \ | \ f(x,y) \Longleftrightarrow f(by+c,y)=0 \in K[y]$$
where $K$ is a field. However, I was unable to find a proof, and it seems strange to me that this fact should be true, because one of the proofs I know for the standard factor theorem (in one variable) uses that $K[x]$ is a Euclidean domain, but $K[x,y]$ is not. So here are my questions:
- Does anybody know a proof / counterexample to the above statement?
- If it is true, are there further generalizations? For instance with more variables or with factors with degree higher than linear or with $K$ not a field.
Edit: The "only if" implication is clear even if instead of $by+c$ we have any polynomial in any number of variables (different from $x$), and also for any ring $K$. The difficult implication is the "if" part, because we need some form of Euclidean division, and it is not clear to me when we have it.