maybe that's a pointless question, however I'm having problems in "understanding" (accepting) the Hilbert's Nullstellensatz. I understand the proof, however I cannot 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 follows directly from the 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?
Thanks in advance.
Best Answer
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.
Ofcourse, @mbrown's comment of rephrasing the Nullstellensatz is perhaps the best way to think about it. Good luck!