Why is there no formula for the roots of a fifth (or higher) degree polynomial equation in terms of the coefficients of the polynomial, using only the usual algebraic operations (addition, subtraction, multiplication, division) and application of radicals (square roots, cube roots, etc)?

Which makes me wonder; would there be a formula if you used more machinery that normal stuff and radicals? What does "more than radicals" look like?

3AprilSR1moI think people usually just use “the number is the root of this polynomial” in
and of itself to describe them, which is indeed more than radicals. There
probably are more round about ways to do it, though.

So a thing Galois theory does is explain:

Which makes me wonder; would there be a formula if you used more machinery that normal stuff and radicals? What does "more than radicals" look like?

https://en.wikipedia.org/wiki/Bring_radical