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It is always weird to see a proof that something is impossible, especially when the tools used in the proof have nothing to do(at a first sight) with the original statement of the problem. I know a few such mathematical theorems, which prove that some facts cannot happen in various areas of mathematics like geometry, algebra, analysis. One example is that there isn't a general formula, using only radicals, for solving polynomial equations of degree greater than $5$. The proof of this theorem uses Galois theory, and this tool was invented especially for solving this problem.

What other theorems do you know which prove that certain facts are impossible in mathematics, and in their proofs something unexpected is used?

Since I was asked to be more specific, I'll tell you more about what I was thinking. I see that a first answer, about the impossibility of the trisection of the angle was posted. This is exactly the kind of answers I have in mind. It is proved that an angle cannot be split into three equal parts using a straightedge and a compass, but the proof has nothing to do with geometry. It uses Galois theory.

Another example: It is impossible to dissect the unit square into an odd number of triangles with equal areas. This is known as Monsky's Theorem, and the only known proof(as far as I know) uses $p$-adic valuations arguments, again, something that we normally wouldn't expect.

I found myself some examples:

Impossibility of making some construction with ruler and compass.

Impossibility of calculating $\int e^{x^2}dx$ in a reasonable closed form. ( I first heard of this one when I was in my first year of college, but the teacher didn't say who proved it, or how it can be done.)

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The unsolvability of the quintic was actually proven before Galois did his work, by Abel and Ruffini. I don't know what their method was. Also, the question strikes me as a little broad; any statement of the form $\forall x, P(x)$ can be phrased as the statement that something is impossible, since it is logically equivalent to the impossibility of finding $x$ such that $\neg P(x)$. Can you be more specific? –  Qiaochu Yuan Mar 12 '12 at 23:17
Hmm, I think that most theorems can be reformulated into "it is impossible to find a counterexample to ...". –  dtldarek Mar 13 '12 at 20:03
@dtldarek: Please read more carefully the question. It clearly says that the proof of impossibility is done by using some tool that is not related to the problem. Something that is out of the ordinary, like the examples given, and the answers received. –  Beni Bogosel Mar 13 '12 at 22:56
@BeniBogosel I did read the question carefully. What I meant was that having extraordinary proof that uses unusual tools, probably you can transform it into an impossibility result. Isn't that what you were asking about? –  dtldarek Mar 13 '12 at 23:20
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5 Answers

A natural impossibility question in knot theory (perhaps the most basic and important one) is to ask whether it is possible, given two knot diagrams, to transform one of them into the other by means of the Reidemeister moves. By Reidemeister's theorem, this is equivalent to them presenting the same knot. Of course if it is possible then we can just exhibit a sequence of moves, but if it is impossible, how to prove this is not so clear.

One method is to construct knot invariants. These are objects such as polynomials constructed from knot diagrams and invariant under the Reidemeister moves; thus, if two knot diagrams have different invariants, they cannot present the same knot. A simple example is tricolorability, which already proves that the trefoil knot is not the unknot. More sophisticated and powerful examples such as the Alexander polynomial and the knot group can be constructed using topological methods (but using the knot group is not as easy as using polynomials because it is undecidable in general whether two groups are isomorphic given only their presentations whereas it is fairly straightforward to decide whether two polynomials are equal).

In 1984 Vaughan Jones discovered the Jones polynomial in a fairly unexpected way while studying operator algebras. This discovery pioneered the now very active field of quantum knot invariants, which has ties to many other fields of mathematics. I don't know a great introductory survey on these ideas but you can consult, for example, Turaev's Quantum invariants of knots and 3-manifolds, and there are also some good keywords and links at the nLab page about knot invariants.

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The Bolyai-Gerwien theorem states that two polygons in $\mathbb{R}^2$ have the same area if and only if it is possible to cut one up into finitely many polygonal pieces and rotate and translate these pieces to obtain the other (scissors congruence). It is natural to ask whether the same is true in higher dimensions, and this is the subject of (a problem closely related to) Hilbert's third problem.

Dehn showed that the answer was no for polyhedrons in $\mathbb{R}^3$. The proof depends on the definition of a nontrivial invariant, the Dehn invariant (see the Wikipedia article for details), of polyhedra that is preserved by scissors congruence. The definition requires the perhaps unexpected use of the $\mathbb{Q}$-vector space $\mathbb{R} / \mathbb{Q} \pi$ as well as the perhaps unexpected use of a tensor product. One then exhibits a pair of polyhedra with the same volume but different Dehn invariants; it is therefore impossible to cut one of them up into pieces and rotate and translate them to make the other.

The structure of this argument is formally similar to the modern Galois-theoretic proof of the unsolvability of the quintic. In modern language, one defines a class of Galois extensions, the solvable extensions, and asks whether an extension defined by the roots of an irreducible quintic is always of that form. The answer is no, and the reason is that one can associate an invariant, the Galois group, to a Galois extension such that there exists an irreducible quintic such that the associated Galois group is not solvable (historically the reason for this name is that the Galois groups of solvable extensions are solvable).

So a general strategy for writing down impossibility proofs in this vein is to find nontrivial sources of invariants, which of course may come from many unexpected areas of mathematics.

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Computability theory/Logic is full of these kinds of statements.

Turing's paper that there are real numbers that cannot be computed by a machine is, I think, a possibility. The (special) proof (technique) is by diagonalization. One can see that there are countably many "recursive functions" hence countably many programs. It is then easy to find, using diagonalization, a real number which cannot be computed by any program.

Another is the impossibility of an algorithm which can decide any statement about mathematics. The original proof, given by Turing, is actually a corollary of the above. A later proof, I believe mistakenly attributed to Turing, is the use of the specialized technique called the halting problem (which is essentially proof by contradiction).

There are also, in computability theory, the so called incompleteness theorems of Gödel. These state that it is impossible to obtain a consistent and complete theory of mathematics.

There are also Cohen's theorems which state that it is impossible to know whether the continuum hypothesis (CH) is True of False. This is because one can construct model's of set theory in which the CH is true and other models in which CH is false. This one uses a great specialized tool called forcing.

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The proof that some elementary functions do not have an elementary integral by Liouville and later generalized by Rosenlicht and others. This latter development introduced Galois theory in that context, which may seem at first extraneous. See this answer for references.

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Something I was discussing just today: it is impossible to trisect the angle (in general). This uses Galois theory as well, albeit to a minimal extent. The key to proving this is to observe that everything in $\mathbb K$, the field of constructable numbers, is made by solving systems of linear and quadratic equations (which come from lines and circles) so any $x\in \mathbb K$ is in $\mathbb Q(x_1)\cdots(x_n)$ for some $x_1,\ldots,x_n$ such that $\mathbb Q(x_1)\cdots(x_k)(x_{k+1})/\mathbb Q(x_1)\cdots(x_k)$ is a quadratic field extension for each $k\in\{0,\ldots,n-1\}$. But the minimal polynomial for $\sin 10^{\circ}$ is $8x^3-6x+1$ (as can be derived using trig identities) and so $\sin 10^{\circ}$ lies in a cubic extension of $\mathbb Q$, hence is not constructable. Since $\sin 30^{\circ}=\frac{1}{2}$, this means $30^{\circ}$ cannot be trisected with a compass and straightedge.

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