Insolvability of the quintic without Galois theory

The explicit goal of most courses on Galois theory is to prove that there exist polynomials whose roots cannot be expressed in termsof a finite combination of rational numbers, the four field operations, and radicals. In the 1960s, V. I. Arnold discovered a much simpler proof of this which barely uses any abstract algebra, and certainly no Galois theory. This approach is not as well-known as it deserves to be. The goal of my talk will be to explain Arnold's ideas, and how the resulting theorem is simultaneously stronger and weaker than what Galois theory gives. Although very little technical background will be required to follow the talk, some amount of mathematical maturity is necessary.