Summary: HILBERT'S NULLSTELLENSATZ
This is the simplest proof of the Nullstellensatz that I have been able to
come up with. It is meant for students learning commutative algebra
for the first time--students perhaps lost in the sea of new vocabulary,
with no clear guidance about which concepts are all-important (e.g.,
Noetherianness, and integrality and finiteness of ring extensions) and
which are less so. Accordingly, we use nothing beyond unique factoriza-
tion in one-variable polynomial rings and the basics of field extensions.
Dan Bernstein led me to some references, and it turns out that my
proof is the same in its essentials as one by Zariski . Zariski's proof
led to the definition of a class of rings called either Jacobson rings or
Hilbert rings, which are defined as "the class of rings to which this
argument applies"; see  for a discussion. Also, our arguments about
denominators motivate the definition of a finite extension of rings, al-
though we avoid using this language explicitly.
I am also grateful to Keith Conrad for his helpful comments.
Theorem. Let k be a field and K a field extension which is finitely
generated as a k-algebra. Then K is algebraic over k.
Example of Proof. Suppose k is infinite and K is the simple tran-