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I read that Hilbert used the nullstellensatz in algebraic number theory rather than in algebraic geometry. What did he use it for? Today, is the nullstellensatz used in algebraic number theory or other areas apart from algebraic geometry?

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    $\begingroup$ Why not tell us where you read that, for some more context? $\endgroup$
    – KConrad
    Commented Mar 15, 2011 at 5:47
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    $\begingroup$ Isn't this a false dichotomy? $\endgroup$
    – Yemon Choi
    Commented Mar 15, 2011 at 9:46
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    $\begingroup$ More to the point: what do you think "algebraic number theory" is, and why do you think it is distinct from "algebraic geometry"? And if you want to know what Hilbert used the Nulstellensatz for, what kind of answer are you hoping for? What level of technicality would be more useful or less useful to you, O Creature of Inconstant Name? $\endgroup$
    – Yemon Choi
    Commented Mar 15, 2011 at 9:50
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    $\begingroup$ "What did he use it for?" He did it for the lulz. $\endgroup$
    – darij grinberg
    Commented Mar 15, 2011 at 9:53
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    $\begingroup$ BTW: I really believe that you might not get so many questions closed, or at least get constructive comments, if you eschewed messing around with different monikers and were honest and open about your own mathematical background (cf. your answer/comment here: mathoverflow.net/questions/20386/mathematics-as-a-hobby/… ) If you ask questions about reasonably technical things, people are going to assume you have the relevant technical background and pitch their answers accordingly. $\endgroup$
    – Yemon Choi
    Commented Mar 15, 2011 at 10:09

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I answer: 'is the Nullstellensatz used in other areas?'

Yes, for example, a variant is used in Combinatorics.

There is a Combinatorial Nullstellensatz (due to N. Alon). The rough idea is to translate certain combinatorial problems into statements on the (non-)vanishing of systems of polynomial equations, and then to analyse this system of polynomial equations.

A question related to it was somewhat recently asked on this site by gowers where you can find details on it.

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