In particular, I am wondering:
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?
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?