# Intuition behind Hilbert's Nullstellensatz

Maybe that's a pointless question, however I'm having problems in "understanding" (accepting) the Hilbert's Nullstellensatz. I understand the proof; however, I do not understand the concept in a more constructive way. I think the source of my doubt is in the fact that maximal ideals in $$k[x_1, x_2, ..., x_n]$$ are always in the form $$\mathfrak{m} = (x_1 - a_1, x_2 - a_2, ..., x_n - a_n )$$ and in the fact that $$k[x_1, x_2, ..., x_n]/\mathfrak{m} \cong k$$ if $$k$$ is algebraically closed. I know that these results follow directly from this result:

If $$R$$ is a finitely generated $$k$$-algebra ($$k$$ maybe not algebraically closed) and $$R$$ is a field, then $$R/k$$ is an algebraic extension

(which I think that's anti-intuitive too and all the proofs that I found seems very unnatural).

Is there any constructive proof or algorithm to these facts or some illuminating example?

• I think Hilbert had $k=\mathbb{C}$ in mind, if that is helpful. Also, another one of the uncountably many ways to phrase the Nullstellensatz is that, if $I \subset k[x_1, \dots, x_n]$ is a proper ideal, where $k$ is algebraically closed, then $V(I) \ne \emptyset$. So, if we take $k=\mathbb{C}$ and $n=1$, this is just the fundamental theorem of algebra, and hence Hilbert's Nullstellensatz is a somewhat natural generalization of a very well-known theorem.
– user55407
Apr 20, 2013 at 1:34

May be you will find this method unnatural as well, but go to The Stacks Project, browse to the chapter "Exercises" (this is chapter $74$), and take a look at exercise $10.1$. You can also look at Spectrum of a linear operator on a vector space of countable dim in which I ask a question related to $10.1$. Note that this proves the Nullstellensatz only for $\mathbb{C}$, but has the advantage of using the language of linear algebra which you may prefer more/ find more intuitive.
The other option would be to convince yourself that Noether Normalization is saying something geometric (the Nullstellensatz is an easy consequence of Noether Normalization), and for this I can recommend Ravi Vakil's "Foundations of Algebraic Geometry" (found here) sections $11.2.3$ to $11.2.6$ in the March 23rd version of the notes, although this may be an overkill.