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Mathematics is rife with the fruit of abstraction. Many problems which first are solved via "direct" methods (long and difficult calculations, tricky estimates, and gritty technical theorems) later turn out to follow beautifully from basic properties of simple devices, though it often takes some work to set up the new machinery. I would like to hear about some examples of problems which were originally solved using arduous direct techniques, but were later found to be corollaries of more sophisticated results.

I am not as interested in problems which motivated the development of complex machinery that eventually solved them, such as the Poincare conjecture in dimension five or higher (which motivated the development of surgery theory) or the Weil conjectures (which motivated the development of l-adic and other cohomology theories). I would also prefer results which really did have difficult solutions before the quick proofs were found. Finally, I insist that the proofs really be quick (it should be possible to explain it in a few sentences granting the machinery on which it depends) but certainly not necessarily easy (i.e. it is fine if the machinery is extremely difficult to construct).

In summary, I'm looking for results that everyone thought was really hard but which turned out to be almost trivial (or at least natural) when looked at in the right way. I'll post an answer which gives what I would consider to be an example.

I decided to make this a community wiki, and I think the usual "one example per answer" guideline makes sense here.

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31 Answers 31

People say that Hilbert's basis theorem was once proven using pages of explicit computation with polynomials, but now everyone learns Hilbert's beautiful, if non-constructive, proof instead. Regrettably I have no idea what the "old proof" looks like.

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What? I think there is by definition no constructive proof of Hilbert's basis theorem. Maybe you mean the Gröbner basis for the invariant ring of a group action, or the projective resolution? These are both still best proven constructively, as the constructive proofs yield tons of additional results. – darij grinberg May 16 '10 at 20:16
Okay, that's always the question with the word "constructive". If we have an ideal given by some equations, or even by generators, how much do we know about the set of leading coefficients of elements of this ideal? Not enough to find its generators. But then again, Hilbert's basis theorem is not 100% constructive itself, for this very reason: we have no idea how the ideal is given. – darij grinberg May 16 '10 at 20:37
@Qiaochu: I read an article on the history of the HBT, and the way it actually "went down" was that the theorem was proven individually for explicit rings. Hilbert proved the HBT and in doing so proved the general theorem. – Harry Gindi May 16 '10 at 22:50
Do read McLarty's article on Gordan's attitude towards Hilbert's work. Historical reality is much more interesting than the myth. – David Corfield May 17 '10 at 13:39
The "old proof" refers to the finite generation of the ring of invariants (for the case of SL(2) acting of binary forms), which is a different theorem. Hilbert found an encompassing approach based on Noether condition for ideals (in fact, he proved much more, in particular, the finiteness of the chain of syzygies); note, however, that in order to apply his "theological" proof to the original problem for more general invariant rings, one still needs to know complete reducibility of representations, which was first established by Hurwitz and Schur using topological arguments (compactness). – Victor Protsak May 18 '10 at 2:22

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