Let $\mathbb{F}$ be an arbitrary field and consider a polynomial of **degree one**: $$f(x_1,\ldots,x_n)=a_1x_1+\cdots+a_nx_n+b\in \mathbb{F}[x_1,\ldots,x_n].$$
Let $H:f=0$ be the corresponding affine hyperplane and let $g\in\mathbb{F}[x_1,\ldots,x_n]$ be any polynomial that vanishes on $H$. If $\mathbb{F}$ is algebraically closed then the classical Nullstellensatz tells us that $f$ divides $g$.

Question: Do we really need algebraic closure in this case?