The big picture is the following: There is a certain kind of groups called "solvable groups". There is a theorem stating that a polynomial equation can be solv

The big picture is the following: There is a certain kind of groups called "solvable groups". There is a theorem stating that a polynomial equation can be solved by radicals if and only if its Galois group is a solvable group. And whether a given finite group (such as a Galois group) is solvable is an entirely finite combinatorial question which can be mechanically checked. Even better: A witness for the solvability of the Galois group can be transformed -- again nontrivially but mechanically -- into explicit formulas for the solutions of the polynomial equation. Wikipedia has some examples on solvable groups and their connection to Galois theory, though I hope that someone has written some blog post or similar explaining this beautiful connection with less prerequisites. https://en.wikipedia.org/wiki/Solvable_group https://en.wikipedia.org/wiki/Solvable_group