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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!

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3  
Just to put it out there, the obvious example is the fundamental theorem of algebra. –  Git Gud Jan 26 at 18:30
    
@GitGud Thanks, you are right! I forgot to mention it, even though it might be the first example of this sort that one sees. –  Sasha Patotski Jan 26 at 18:32
3  
Many impossibility proofs are of this kind: In order to prove that all constructions in the framework of the obvious theory (e.g., constructions with ruler and compass, or derivation of elementary functions) cannot solve a particular problem you need a "transcending" theory. –  Christian Blatter Jan 26 at 18:43
    
Recently I woked on a result which relates information theory and group theory, and whose proof relies partly on topological arguments in $\mathbf{R}^n$. Sadly, the original article is behind a paywall : ieeexplore.ieee.org/iel5/18/21818/01013138.pdf –  fkraiem Jan 26 at 19:23
    
The Poincaré conjecture. –  Martín-Blas Pérez Pinilla Jan 26 at 19:35

2 Answers 2

up vote 7 down vote accepted
  1. In number theory, to prove the sharpest bounds on character sums (such as $\sum_{a=1}^{p-1} (\frac{a^3-2}{p})$, where $(\frac{\cdot}{p})$ is the Legendre symbol) we use algebraic geometry: interpret the character sum as the linear coefficient in the zeta-function of a curve or higher-dimensional variety over a finite field and use the Riemann hypothesis for varieties over finite fields.

  2. The Ramanujan-Petersson conjecture on bounds on coefficients of the modular discriminant function $\Delta(z)$ is at first a statement in the setting of complex analysis, but its proof requires an algebro-geometric interpretation of the coefficients you are trying to bound.

  3. To prove there are infinitely many primes $p \equiv 2 \bmod 5$, which is at first a statement just about prime numbers, we use (complex) analysis to prove Dirichlet's theorem on primes in arithmetic progression. Some other arithmetic progressions, such as $p \equiv 1 \bmod 4$, can be proved infinite in size by purely algebraic techniques, but I don't believe this has ever been done for $p \equiv 2 \bmod 5$.

  4. That every positive integer is a sum of four squares is a statement purely about integers, but it has proofs that bring in ideas from other areas of math. There is a proof using division with remainder in the quaternions, there is a proof using Minkowski's geometry of numbers (which is considered a part of number theory because that is where its principal applications lie, but on first glance it is unexpected to use properties of volumes in Euclidean space to draw conclusions about integers), and there is a proof by Jacobi using analysis (modular forms).

  5. The Ax-Kochen theorem is about zeros of homogeneous polynomials over $p$-adic fields and its proof uses mathematical logic. The point is that an analogous property of homogeneous polynomials was proved earlier over all the Laurent series fields ${\mathbf F}_p((x))$, and any nonprincipal ultraproduct of the fields ${\mathbf F}_p((x))$ over all primes $p$ is elementarily equivalent to a nonprincipal ultraproduct of the field ${\mathbf Q}_p$ over all primes $p$. Then model theory lets one make conclusions about a property in all but finitely many fields ${\mathbf Q}_p$ from knowledge of the property in all (or all but finitely many) of the fields ${\mathbf F}_p((x))$.

  6. In harmonic analysis, Wiener proved that the reciprocal of a nonvanishing absolutely convergent Fourier series also has an absolutely convergent Fourier series. His proof was apparently very complicated. I never read it. Later Gelfand introduced algebraic techniques (maximal ideals in Banach algebras) to give a more conceptual proof.

  7. To derive a formula for the $n$-th Fibonacci number, you could use linear algebra (study all sequences satisfying the same recursion as the Fibonacci numbers and write the Fibonacci sequence in terms of a concrete basis for such sequences) or analysis (formal power series, a.k.a. generating functions). In a sense, the linear nature of so many problems in math is what makes linear algebra such an important tool.

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One nice example is Kneser conjecture

whenever the $n$-subsets of a $2n+k$-set are divided into $k+1$ classes, then two disjoint subsets end up in the same class. The short and nice proof is given by arranging the points on the sphere and using Borsuk-Ulam theorem e.g. by topology!

Also topological notions like Schwartz genus are used in complexity theory (some of the obstacles in solving polynomial equations are purely topological). You can find a lot of examples in Vasiliev's Topology of complement's of discriminats

There are a lot of beautiful "moduli space" arguments in algebraic geometry, this is not from completely different field, but proving 27 lines on the cubic (very coarse theorem ) by computing chern classes of canonical bundle on $2-4$ Grassmanian (purely topological object) is interesting

There are a lot of applications of linear algebra to combinatorial constructions e.g. the counterexample to Borsuk conjecture or Frankl-Wilson theorem

Another nice example are given by standart geometric proofs of finiteness of ideal class group.

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