Let be $k$ some field, where I have $J$ maximal ideal of $k[x, y] / (f)$ for some $f \in k[x, y]$.
I want to know if it's possible for $J$ to be embedded as a maximal ideal of $k'[x, y] / (f)$.
For the context, I have some affine curve $K$ and I'm looking at the map $P = (a, b) \mapsto (x - a, y - b)$, this map is injective, but in the case where the field is algebraically closed, it becomes bijective.
I already know that (weak) Nullenstellensatz gives me some interesting results between maximal ideals and points of a curve, but it only works for algebraically closed fields. So I'm struggling and don't know if I miss (or misunderstood) something
Let $J$ be a maximal ideal of $k[x,y]$ containing $f$,
The main point is that $k[x,y]/J$ is algebraic over $k$. This is because neither $k(t)$ nor $k(t,s)$ nor their finite extensions are finitely generated as a $k$-algebra.
Thus $J = (g(x),h(x,y))$ where $g(x)\in k[x]$ is non-zero irreducible and $h(x,y)\in k[x]/(g(x))[y]$ is non-zero irreducible.
The maximal ideals of $\overline{k}[x,y]$ containing $J$ are those of the form $(x-a,y-b)$ where $a\in \overline{k}$ is a root of $g(x)\in k[x]$ and $b\in \overline{k}$ is a root of $h(a,y)\in k(a)[y]$. If $k$ has characteristic $0$ then there are $\deg(g)\deg(h) = [k[x,y]/J:k]$ of them and $\overline{k}[x,y]/J\cong \overline{k}^{[k[x,y]/J:k]}$