Why can some polynomial equations not be solved by a formula using radicals?

In modern algebra, the inability to solve certain polynomial equations using radicals—meaning formulas involving only addition, subtraction, multiplication, division, and root extraction—is determined by the properties of the equation's associated group of permutations of its roots.123

While formulas for quadratic, cubic, and quartic equations were discovered in the 16th century, mathematicians long struggled to find a general formula for quintic (fifth-degree) and higher-degree equations. In 1824, Niels Henrik Abel provided the first valid proof that a general solution by radicals is impossible for equations beyond the fourth degree.314

Building on this, Évariste Galois developed a theory showing that a polynomial equation is solvable by radicals if and only if its associated permutation group is "soluble". A group is considered soluble if it possesses a specific structure, characterized by a chain of normal subgroups with cyclic quotients. When this condition is not met, no radical formula can exist, marking a transition in mathematics from formal calculation to the conceptual analysis of group theory.23