Mathematics
Algebraic Varieties and the Nullstellensatz
Quick fact
Hilbert's Nullstellensatz (German for 'theorem of zeros') states that a polynomial vanishes on an algebraic variety if and only if some power of it lies in the ideal that defines the variety. This bridges geometry and algebra in a way that allows geometric questions to be solved algebraically.
Why this is interesting
You've likely graphed a circle or a parabola. But what if you consider all points that satisfy several polynomial equations at once? That's an algebraic variety. And there's a stunning theorem that tells you exactly when a polynomial vanishes on such a set.