Why did Hilbert prove the Nullstellensatz?

Question

In particular, I am wondering:

1. Historically, what was the impetus for the development of the connection between the algebra of polynomial rings and the geometry of algebraic varieties? Concretely, what problems were solved by Hilbert's work?

2. From a modern perspective, what are the most illustrative motivating examples which demonstrate the utility of the Nullstellensatz for studying the geometry of polynomial curves?

I am studying elementary algebraic geometry from a variety of introductory texts, and I find the Nullstellensatz to be a very deep theorem, but either I'm missing something, or these texts are not actually telling me what you can do with the Nullstellensatz which is of direct relevance to the study of particular algebraic varieties! A very compelling dictionary between algebra and geometry is created, but can someone please give me an example of where this dictionary is used to say something interesting about geometry?

Answer 1

Here are two elementary and important consequences/rephrasements of HNS. They should be included in whatever textbook you are using.

1. Every maximal ideal in $k[x_{1},...,x_{n}]$ (with $\bar{k}=k$) has the form $$ (x_{1}-a_{1},...,x_{n}-a_{n}) $$ for some $a_{1},...,a_{n}\in k$.

2. There is a one-to-one correspondence between radical ideals in $k[x_{1},...,x_{n}]$ (with $\bar{k}=k$) and affine varieties in $k^{n}$. Moreover, the ideal is prime if and only if the variety is irreducible.

I think that the importance of these two results should be pretty clear even at first glance. In particular the second one is especially important (in my opinion). It is in the very heart of algebraic geometry: what we use to go from algebra to geometry and viceversa is the equivalence of categories (reversing arrows) between affine varieties and their coordinate rings (reduced finitely generated $k$-algebras).

In fact, the introduction of schemes in algebraic geometry generalizes this equivalence in a vast way: affine schemes (with arrows reversed) are equivalent to commutative rings as categories.

Answer 2

Which polynomials $P(X,Y)\in \mathbb C[X,Y]$ vanish on every point in $\mathbb C^2$ of the complex circle $X^2+Y^2-1=0$ ?
Well there is $P(X,Y)=X^2+Y^2-1$, duh!
Is that all? No: any polynomial of the form $P(X,Y)=Q(X,Y)(X^2+Y^2-1)$ where $Q(X,Y)$ is a completely arbitrary polynomial obviously also vanishes on all points of our circle.
Is that all? $$\operatorname {YES \;!!}$$ And this (suitably generalized) is the content of the Nullstellensatz!