I would like to know more examples of theorems, which "belong to one field of math", but their proofs are from the "outside of the field".
I am mostly interested in proofs that are not too long (not like the proof of Fermat's last theorem), but where the main part of the proof is to guess to look for a solution "outside the current setting" the problem is stated.
Let me give a couple of examples, so that it would be more clear what I am looking for.
For example, any finite field $F$ has $p^k$ elements, for some prime number $p$. To prove this, one should notice that if $p=char(F)$, then $F$ is actually a finite dimensional vector space over $\mathbb{F}_p$. Then the proof is obvious. So here to prove some fact about fields we use linear algebra.
Another example. Every subgroup of a free group $\Gamma$ is free. For this, one notices that $\Gamma$ can be thought as the fundamental group $\pi_1$ of a bouquet of circles. Then subgroups $H$ of $\pi_1=\Gamma$ correspond to coverings. Coverings of a graph are graphs. So $H$ is a fundamental group of a covering, which is a graph, which can be homotoped to a bouquet of circles. So $H$ is also free. Here we used topology.
Also the Brower fixed point theorem is stated in a very elementary topological terms, but the proof (at least the shortest one I know) uses homology.
Another example is the third Hilbert's problem: if we have two polyhedra of the same volume, can we cut one of them into smaller pieces (by straight cuts) and then re-arrange them to obtain the other polyhedron? To answer this question, one needs an invariant called Dehn invariant, which is an element of $\mathbb{R}\otimes_{\mathbb{Q}}\mathbb{R}/\mathbb{Q}$ (see for example these notes, Prob.1.51). Here we solved a problem about classical euclidian geometry using some linear algebra.
I am looking for more examples, but maybe not as simple as the first three examples I gave. At the same time, I want examples that can be explained (maybe omitting details) to a general grad student in about 20-30 minutes.
I hope I am not being too picky here))
Thank you very much for your help!