proof of Hilbert’s Nullstellensatz
Let be an algebraically closed field, let , and let be an ideal of the polynomial ring . Let bea polynomial
with the property that
Suppose that for all ; in particular, is strictly smaller than and . Consider the ring
The -ideal is strictly smaller than , since
does not contain the unit element. Let be an indeterminate over, and let be the inverse image of underthe homomorphism
acting as the identity on and sending to. Then is strictly smaller than , sothe weak Nullstellensatz gives us an element such that for all . Inparticular, we see that for all . Ourassumption
on therefore implies . However, also contains the element since sends this elementto zero. This leads to the following contradiction
:
The assumption that for all is therefore false,i.e. there is an with .