I'm a graduate student who's been learning about schemes this year from the usual sources (e.g. Hartshorne, Eisenbud-Harris, Ravi Vakil's notes). I'm looking for some examples of elementary self-contained problems that scheme theory answers - ideally something that I could explain to a fellow grad student in another field when they ask "What can you do with schemes?"
Let me give an example of what I'm looking for: In finite group theory, a well known theorem of Burnside's is that a group of order $p^a q^b$ is solvable. It turns out an easy way to prove this theorem is by using fairly basic character theory (a later proof using only 'elementary' group theory is now known, but is much more intricate). Then, if another graduate student asks me "What can you do with character theory?", I can give them this example, even if they don't know what a character is.
Moreover, the statement of Burnside's theorem doesn't depend on character theory, and so this is also an example of character theory proving something external (e.g. character theory isn't just proving theorems about character theory).
I'm very interested in learning about similar examples from scheme theory.
What are some elementary problems (ideally not depending on schemes) that have nice proofs using schemes?
Please note that I'm not asking for large-scale justification of scheme theoretic algebraic geometry (e.g. studying the Weil conjectures, etc). The goal is to be able to give some concrete notion of what you can do with schemes to, say, a beginning graduate student or someone not studying algebraic geometry.