solution by radicals

Solution by radicals is a concept in algebra concerning the ability to solve polynomial equations using only basic arithmetic operations and root extractions. For centuries, mathematicians sought general formulas to express the roots of polynomial equations in terms of their coefficients. While solutions were found for quadratic, cubic, and quartic equations, attempts to find a general solution for quintic (fifth-degree) equations and higher proved unsuccessful.12

In the early 19th century, mathematicians like Paolo Ruffini and Niels Henrik Abel proved that a general solution by radicals for polynomial equations of degree five or higher is impossible. Évariste Galois later developed group theory, which provided the precise conditions under which a polynomial equation can be solved by radicals. This work established that an equation is solvable by radicals if and only if its associated permutation group is "solvable".134