TITLE: List of Mathematical Impossibilities proved using special tools
QUESTION [12 upvotes]: 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.)

REPLY [6 votes]: 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.