From what I've studied, Abel-Ruffini theorem states that we can't find all the roots of some polynomials with degree above 5, using only radicals and arithmetic operations.
How does it imply that we can't have a formula that involves some other mathematics thus allowing us to find all the roots of any polynomial? Do abstract algebra and Galois theory have a role in this I'm not aware of?
It doesn't imply there isn't some other formula. Hermite, for example, found a solution for the general quintic using theta functions.