Nullstellensatz corollary: maximal ideals are points

A key consequence of the Nullstellensatz variety correspondence is that maximal ideals in a polynomial ring over an algebraically closed field have an explicit and geometric form.

Corollary (Weak Nullstellensatz / maximal ideals of $k[x_1,\dots,x_n]$)

Let $k$ be an algebraically closed field . If $\mathfrak m \subset k[x_1,\dots,x_n]$ is a maximal ideal, then there exists a point $a=(a_1,\dots,a_n) \in k^n$ such that

Equivalently, the maximal spectrum of $k[x_1,\dots,x_n]$ is naturally identified with the set of $k$-rational points $k^n$, and the residue field at $\mathfrak m$ is canonically $k$.

Under this identification, the subspace topology induced from the Zariski topology on Spec agrees with the usual “vanishing set” Zariski topology on $k^n$.

Examples

1. One variable.
   In $\mathbb C[x]$, every maximal ideal is of the form $(x-a)$ for a unique $a \in \mathbb C$.

2. Two variables.
   In $\mathbb C[x,y]$, the ideal $(x-1,\; y+2)$ is maximal and corresponds to the point $(1,-2) \in \mathbb C^2$.

3. A non-maximal ideal and the points above it.
   In $\mathbb C[x]$, the ideal $(x^2+1)$ is not maximal because $x^2+1=(x-i)(x+i)$. The maximal ideals containing $(x^2+1)$ are $(x-i)$ and $(x+i)$, corresponding to the two points $i$ and $-i$.