Find $k$ s.t. $\mathbb{A}^{2}_{k}\to \mathbb{A}^{1}_{k}$ is not closed.

Question

Let $k$ be a field and let $f:\mathbb{A}^{2}_{k}\to \mathbb{A}^{1}_{k}$ be a map between affine scheme induced by natural inclusion $i:k[Y]\hookrightarrow k[X, Y]$, $Y\mapsto Y$. I want to show that this map is not a closed map in general, but I don't know how to find such counterexample. First, I tried to find a maximal ideal $\mathfrak{m}\subset k[X, Y]$ s.t. $i^{-1}(\mathfrak{m})=\mathfrak{m}\cap k[Y]\subset k[Y]$ isn't a maximal ideal, which shows that $f$ maps the closed point $\mathfrak{m}_{x}\in \mathbb{A}^{2}_{k}$ to some non-closed point $i^{-1}(\mathfrak{m})_{x}$. However, there's no such example if $k$ is algebraically closed by Hilbert's Nullstellensatz. So I tried to find such example when $k=\mathbb{R}$ or $k=\mathbb{F}_{p}$, but I don't know well how maximal ideals of $k[X, Y]$ looks like in these cases.