It is well-known that if $k$ is algebraically closed field that has infinite transcendence degree over the prime field $\mathbb{Q}$ or $\mathbb{F}_p$ then the maximal ideals of $k[x_1,...,x_n]$ are of the form $(x_1-\alpha_1,...,x_n-\alpha_n)$ for some $\alpha_j\in k$.
Are the assumptions for $k$ necessary ? (Of course, they are sufficient for the standard proof). Can we take weaker condition on $k$ ? I'm looking for some generalization of this theorem.
Is there any characterization of fields for which Hilberts's Nullstellensatz is valid ?
Thanks in advance for any articles and other sources where I can find such information.
The "infinite transcendence degree" part is useless, I don't know where you saw that.
On the other hand, the Nullstellensatz characterizes algebraically closed fields : if $k$ is not algebraically closed, then there is a irreducible polynomial $P\in k[X]$ of degree not $1$, so $(P)$ is a maximal ideal in $k[X]$ not of the form $(X-a)$, so the Nullstellensatz is false (already for $n=1$).