Nullstellensatz corollary: maximal ideals are points

Corollary (Nullstellensatz: maximal ideals correspond to points).
Let $k$ be an algebraically closed field and let $R=k[x_1,\dots,x_n]$. Then every maximal ideal $\mathfrak m\subset R$ is of the form

for a unique point $a=(a_1,\dots,a_n)\in k^n$. Equivalently,

where $\mathrm{ev}_a$ is evaluation at $a$.

This is a standard corollary of the (weak) Hilbert Nullstellensatz , and it underlies the variety–ideal correspondence .

Related knowls.

• Weak Nullstellensatz
• Maximal ideal
• Residue field

Examples

1. One variable over $\mathbb{C}$.
   In $\mathbb{C}[x]$, the maximal ideals are exactly $(x-a)$ with $a\in\mathbb{C}$. The quotient $\mathbb{C}[x]/(x-a)\cong\mathbb{C}$.

2. A point in $\mathbb{C}^2$.
   In $\mathbb{C}[x,y]$, the ideal

   $$(x-1,\;y-i)$$

   is maximal and corresponds to the point $(1,i)\in\mathbb{C}^2$. Indeed, $\mathbb{C}[x,y]/(x-1,y-i)\cong\mathbb{C}$.

3. Why “algebraically closed” matters.
   Over $\mathbb{R}$, the ideal $(x^2+1)\subset\mathbb{R}[x]$ is maximal, but it is not of the form $(x-a)$ for any $a\in\mathbb{R}$. (Its quotient is $\mathbb{R}[x]/(x^2+1)\cong\mathbb{C}$.)