How does a Galois group help determine whether an equation can be solved using radicals?

In modern algebra, the Galois group of a polynomial equation is the group of all admissible permutations of its roots that preserve the algebraic structure of the equation.12

Évariste Galois demonstrated that a polynomial equation is solvable by radicals—meaning its roots can be expressed using only addition, subtraction, multiplication, division, and the extraction of roots—if and only if its associated Galois group is "solvable".314

A group is considered solvable if it can be decomposed into a chain of subgroups where the quotients are cyclic. Essentially, this means the group can be broken down into simpler, "prime-order" constituents with a structure that is easily understood. If the Galois group lacks this specific structural property, no formula involving only radicals can exist to solve the equation. This discovery shifted the focus of algebra from attempting to calculate specific radical solutions to analyzing the conceptual structure of the underlying permutation groups.423