I have proven the following theorem:
Let $p(\vec{x}), q(\vec{x})\in\mathbb{F}[\vec{x}]$ be multivariate polynomials in $n$ variables. If $q$ is linear and $q(\vec{x}) = 0$ implies $p(\vec{x}) = 0$ for all $\vec{x} \in \mathbb{F}^n$, then $q$ is a factor of $p$.
I do not know the name of this theorem. I am wondering what theory(ies) or field(s) of mathematics this theorem relates to, and whether this theorem can be generalized.
In particular, the theorem fails if $q$ is only required to be irreducible rather than linear. Take $p = (x^2-1)^2 + (y^2-1)^2$ and $q = (x-1)^2 + (y-1)^2$ with $\mathbb{F} = \mathbb{R}$.
I have tried searching for a while, but could not find much (likely because I do not know what keywords to search for).
Any help is greatly appreciated!
After a bit more searching, I discovered that this theorem is a special case of Bézout's theorem. The generalization to irreducible $q$ is true as long as $p$ and $q$ share infinitely many zeroes.