For a quadratic polynomial there exists a formula for its roots. I read that similarly for polynomials of degree 3 and 4 there also exists such a formula but that no such formulas exist for polynomials of degree 5 or higher.
Does there exist a proof that no such formulas for $\ge 5$ can exist or is it that they have not been found yet?
On
There is a proof, and it is quite beautiful. The result is known as the Abel-Ruffini Theorem, and the standard proof of this fact uses Galois theory and the fact that the alternating group on $n\ge 5$ symbols is not solvable. This wasn't the original proof, but it is the most elegant. The link provided contains a brief outline of the proof based on Galois's ideas.
There does exists a proof that polynomials with degree greater than 4 cannot be solved with radicals, the proof relies heavily on Galois theory. See for example here.