So a thing Galois theory does is explain:
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?
In light of reading through Raemon's shortform feed, I'm making my own. Here will be smaller ideas that are on my mind.