List of Mathematical Impossibilities proved using special tools 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.)

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.
