Nullstellensatz corollary: maximal ideals are points

Let $k$ be an algebraically closed field. The weak Nullstellensatz states that every maximal ideal $\mathfrak m\subset k[x_1,\dots,x_n]$ has the form

\mathfrak m = (x_1-a_1,\dots,x_n-a_n).

for a unique point $a=(a_1,\dots,a_n)\in k^n$ . Thus the maximal spectrum of $k[x_1,\dots,x_n]$ is naturally identified with $k^n$ , and the residue field at every maximal ideal is canonically $k$ .

Geometric interpretation

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

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

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

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 $-i$ .