Suppose that

is a non-zero ideal of

. Define the variety associated with

as

.

__Claim__: If there exists

such that

then

is a maximal ideal.

__Failed attempt__: For all

from the definition of

. Define the substitution homomorphism

as

. It suffices to prove that

, for then, by Hilbert's Nullstellensatz,

is a maximal ideal.

However, it doesn't seem that

is true. Certainly

. But there may be polynomials in

that are not in

it seems. For

but there could be some

if

and there exists

such that

. It seems this could plausibly occur, but I haven't been able to construct such a counter example. So I am unable to prove that

.

Finally, in working with this problem, I couldn't see a way to get to the structure of

through

. I thought maybe a contrapositive or proof by contradiction might work, but to no avail.