Everybody knows that nobody can solve the quintic. Indeed this insolubility is a well known hard theorem, the high point of a full-semester course on Galois theory, often taken in one's 3rd or 4th year of university mathematics. I'm not sure why so few know that the same theorem can be proven in about 15 minutes using *very* basic and easily understandable topology, accessible to practically anyone.
Prerequisites.
1. The first week of any class on group theory.
2. Knowing that every complex number other than 0 has exactly n roots of order n, and how to compute them.