What are the most overpowered theorems in mathematics?
By “overpowered,” I mean theorems that allow disproportionately strong conclusions to be drawn from minimal / relatively simple assumptions. I’m looking for the biggest guns a research mathematician can wield.
This is different from “proof nukes” (that is, applying advanced results to solve much simpler problems). It is also not the same as requesting very important theorems, since while those are very beautiful and high level and motivate the creation of entire new disciplines of mathematics, they aren’t always commonly used to prove other things (e.g. FLT), and if they are they tend to have more elaborate conditions that are more proportional to the conclusions (e.g. classification of finite simple groups).
Answers should contain the name and statement of the theorem (if it is named), the discipline(s) it comes from, and a brief discussion of why the theorem is so good. I’ll start with an example.
The Feit-Thompson Theorem. All finite groups of odd order are solvable.
Solvability is an incredibly strong condition in that it immediately eliminates all the bizarre, chaotic things that can happen in finite nonabelian simple groups. Solvable groups have terminating central and derived series, a composition series made of cyclic groups of prime order, a full set of Hall subgroups, and so on. The fact that we can read off all that structure simply by looking at the parity of a group’s order is amazing.
I would include the pigeonhole principle in such a list. This principle is one that is easy to explain to a 4 year old, and at the same time it is used throughout all areas of mathematical research. As an example see Proof of Van der Waerden’s theorem (in a special case).