Home » Algebra: Rings

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 kk be an algebraically closed , and let II be a proper of the k[x1,…,xn]k[x_1,\dots,x_n]. Then the affine variety

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

is nonempty.

Equivalently, every of k[x1,…,xn]k[x_1,\dots,x_n] has the form (x1−a1,…,xn−an)(x_1-a_1,\dots,x_n-a_n) for some a=(a1,…,an)∈kna=(a_1,\dots,a_n)\in k^n.