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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.

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    $\begingroup$ 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. $\endgroup$ – user55407 Apr 20 '13 at 1:34
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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!

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  • $\begingroup$ Nice lecture notes. $\endgroup$ – user40276 Apr 24 '13 at 16:51

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