Hilbert's Nullstellensatz (weak)

Over an algebraically closed field, every proper ideal in a polynomial ring has a common zero.

Hilbert's Nullstellensatz (weak)

Hilbert’s Nullstellensatz (weak): Let $k$ be an algebraically closed field , and let $I$ be a proper ideal of the polynomial ring $k[x_1,\dots,x_n]$ . Then the affine variety

V(I)=\{a\in k^n : f(a)=0\text{ for all }f\in I\}

is nonempty.

Equivalently, every maximal ideal of $k[x_1,\dots,x_n]$ has the form $(x_1-a_1,\dots,x_n-a_n)$ for some $a=(a_1,\dots,a_n)\in k^n$ .

Proof sketch: If $\mathfrak m$ is maximal, then $A:=k[x_1,\dots,x_n]/\mathfrak m$ is a finitely generated $k$ -algebra that is a field. The images of the $x_i$ are algebraic over $k$ ; algebraic closedness forces $A\cong k$ , giving $\mathfrak m=(x_1-a_1,\dots,x_n-a_n)$ and hence a common zero.